INTERNATIONAL GONGRESS 44Q996 639 977 612T954 581Q953 580 938 580Q909 582 884 587L869 591L870 587Q870 583 849 557T796 491T748 422Q729 391 692 313T620 188Q555 105 454 34T253 -37Q214 -37 181 -30T120 -2T92 53Q92 89 119 123T184 158Q205 158 215 146T225 119Q225 102 203 89T161 75Q153 75 145 78T135 81Q130 81 130 62Q130 39 153 24T204 5T267 0Q311 0 358 29T454 117T539 226T629 358T710 476Q726 496 744 516T778 551T807 577T828 595L836 601L785 623Q743 642 713 651T668 661T626 663Q564 663 509 644T418 596T356 535T317 475T305 431Q305 416 312 408Q323 388 369 388Q429 388 465 411T530 480Q557 526 557 565Q557 573 556 579T555 587T555 590Q555 591 568 600T584 611Q588 612 600 603Q622 581 622 549Q622 516 600 475T536 405Q454 350 354 350Z">Tcb;
(2) For fixed A A AAA, in one of T c T c T^(c)\mathscr{T}^{c}Tc or T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}Tcb, we have H ( A [ i ] ) = 0 H ( A [ i ] ) = 0 H(A[i])=0H(A[i])=0H(A[i])=0 if i 0 i ≪ 0 i≪0i \ll 0i≪0.
The finite functors are those for which we also have
(3) H ( A [ i ] ) = 0 H ( A [ i ] ) = 0 H(A[i])=0H(A[i])=0H(A[i])=0 for all i 0 i ≫ 0 i≫0i \gg 0i≫0.
Remark 6.8. The proof of part (1) of Theorem 6.6 may be found in [29], while the proof of part (2) of Theorem 6.6 occupies [28]. These are not easy theorems.
Let T = D q c ( X ) T = D q c ( X ) T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)T=Dqc(X), with X X XXX a scheme proper over a noetherian ring R R RRR. Then the hypotheses of Theorem 6.6(1) are satisfied. We learn (among other things) that the natural
1 What's important for the current survey is that, if X X XXX is a noetherian, separated scheme, then T = D q c ( X ) T = D q c ( X ) T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)T=Dqc(X) satisfies this hypothesis provided X X XXX is finite-dimensional and quasiexcellent.
functor, taking an object B D coh b ( X ) B ∈ D coh  b ( X ) B inD_("coh ")^(b)(X)B \in \mathbf{D}_{\text {coh }}^{b}(X)B∈Dcoh b(X) to the R R RRR-linear functor Hom ( , B ) : D perf ( X ) o p Hom ⁡ ( − , B ) : D perf  ( X ) o p → Hom(-,B):D^("perf ")(X)^(op)rarr\operatorname{Hom}(-, B): \mathbf{D}^{\text {perf }}(X)^{\mathrm{op}} \rightarrowHom⁡(−,B):Dperf (X)op→ Mod- R R RRR, is a fully faithful embedding
D coh b ( X ) Y i Hom R ( D p e r f ( X ) o p , R M o d ) D coh  b ( X ) → Y ∘ i Hom R ⁡ D p e r f ( X ) o p , R − M o d D_("coh ")^(b)(X)rarr"Y@i"Hom_(R)(D^(perf)(X)^(op),R-Mod)\mathbf{D}_{\text {coh }}^{b}(X) \xrightarrow{\mathscr{Y} \circ i} \operatorname{Hom}_{R}\left(\mathbf{D}^{\mathrm{perf}}(X)^{\mathrm{op}}, R-\mathrm{Mod}\right)Dcoh b(X)→Y∘iHomR⁡(Dperf(X)op,R−Mod)
whose essential image is precisely the finite homological functors.
If we further assume that the scheme X X XXX is finite-dimensional and quasiexcellent then the hypotheses of Theorem 6.6(2) are also satisfied. We learn that the functor, taking an object A D perf ( X ) A ∈ D perf  ( X ) A inD^("perf ")(X)A \in \mathbf{D}^{\text {perf }}(X)A∈Dperf (X) to the R R RRR-linear functor Hom ( A , ) Hom ⁡ ( A , − ) Hom(A,-)\operatorname{Hom}(A,-)Hom⁡(A,−), is a fully faithful embedding
whose essential image is also the finite homological functors.
In [31, HISTORICAL SURVEY 8.2] the reader can find a discussion of the (algebrogeometric) precursors of Theorem 6.6. As for the applications, let us go through one of them.
Remark 6.9. Let X X XXX be a scheme proper over the field C C C\mathbb{C}C of complex numbers, and let X an X an  X^("an ")X^{\text {an }}Xan  be the underlying complex analytic space. The analytification induces a functor we will call L : D coh b ( X ) D coh b ( X an ) L : D coh  b ( X ) → D coh  b X an  L:D_("coh ")^(b)(X)rarrD_("coh ")^(b)(X^("an "))\mathscr{L}: \mathbf{D}_{\text {coh }}^{b}(X) \rightarrow \mathbf{D}_{\text {coh }}^{b}\left(X^{\text {an }}\right)L:Dcoh b(X)→Dcoh b(Xan ), it is the functor taking a bounded complex of coherent algebraic sheaves on X X XXX to the analytification, which is a bounded complex of coherent analytic sheaves on X an X an  X^("an ")X^{\text {an }}Xan . The pairing sending an object A D perf ( X ) A ∈ D perf  ( X ) A inD^("perf ")(X)A \in \mathbf{D}^{\text {perf }}(X)A∈Dperf (X) and an object B D coh b ( X an ) B ∈ D coh  b X an  B inD_("coh ")^(b)(X^("an "))B \in \mathbf{D}_{\text {coh }}^{b}\left(X^{\text {an }}\right)B∈Dcoh b(Xan ) to Hom ( L ( A ) , B ) Hom ⁡ ( L ( A ) , B ) Hom(L(A),B)\operatorname{Hom}(\mathscr{L}(A), B)Hom⁡(L(A),B) delivers a map
D coh b ( X a n ) Hom R ( D perf ( X ) o p , C M o d ) D coh  b X a n ⟶ Hom R ⁡ D perf  ( X ) o p , C − M o d D_("coh ")^(b)(X^(an))longrightarrowHom_(R)(D^("perf ")(X)^(op),C-Mod)\mathbf{D}_{\text {coh }}^{b}\left(X^{\mathrm{an}}\right) \longrightarrow \operatorname{Hom}_{R}\left(\mathbf{D}^{\text {perf }}(X)^{\mathrm{op}}, \mathbb{C}-\mathrm{Mod}\right)Dcoh b(Xan)⟶HomR⁡(Dperf (X)op,C−Mod)
Since the image lands in the finite homological functors, Theorem 6.6(1) allows us to factor this uniquely through the inclusion Y i Y ∘ i Y@i\mathscr{Y} \circ iY∘i, that is, there exists (up to canonical natural isomorphism) a unique functor R R R\mathscr{R}R rendering commutative the triangle
And proving Serre's GAGA theorem reduces to the easy exercise of showing that L L L\mathscr{L}L and R R R\mathscr{R}R are inverse equivalences, the reader can find this in the (short) [29, SECTION 8 AND APPENDIX A].
The brilliant inspiration underpinning the approach is due to Jack Hall [12], he is the person who came up with the idea of using the pairing above, coupled with representability theorems, to prove GAGA. The representability theorems available to Jack Hall at the time were not powerful enough, and Theorem 6.6 was motivated by trying to find a direct path from the ingenious, simple idea to a fullblown proof.
Discussion 6.10. In preparation for the next theorem, we give a very brief review of metrics in triangulated categories. The reader is referred to the survey article [30] for a much fuller and more thorough account.
Given a triangulated category T T T\mathscr{T}T, a metric on T T T\mathscr{T}T assigns a length to every morphism. In this article the only metrics we consider are the ones arising from t-structures. If T T T\mathscr{T}T is an approximable triangulated category we choose a t-structure ( T 0 , T 0 ) T ≤ 0 , T 0 (T <= 0,T^(0))\left(\mathscr{T} \leq 0, \mathscr{T}^{0}\right)(T≤0,T0) in the preferred equivalence class, and this induces a metric as follows. Given a morphism f : X Y f : X → Y f:X rarr Yf: X \rightarrow Yf:X→Y, we may complete to an exact triangle X f Y D X → f Y → D Xrarr"f"Y rarr DX \xrightarrow{f} Y \rightarrow DX→fY→D, and the length of f f fff is given by the formula
Length ( f ) = inf { 1 2 n | n Z and D T n }  Length  ( f ) = inf 1 2 n n ∈ Z  and  D ∈ T ≤ − n " Length "(f)=i n f{(1)/(2^(n))|n inZ" and "D inT <= -n}\text { Length }(f)=\inf \left\{\left.\frac{1}{2^{n}} \right\rvert\, n \in \mathbb{Z} \text { and } D \in \mathscr{T} \leq-n\right\} Length (f)=inf{12n|n∈Z and D∈T≤−n}
In this survey we allow the length of a morphism to be infinite; if the set on the right is empty then we declare Length ( f ) = ( f ) = ∞ (f)=oo(f)=\infty(f)=∞.
This metric depends on the choice of a t t ttt-structure, but not a lot. As all t t ttt-structures in the preferred equivalence class are equivalent, any two preferred t-structures will give rise to equivalent metrics (with an obvious definition of equivalence of metrics).
Note that if T T T\mathscr{T}T is a triangulated category and S S S\mathscr{S}S is a triangulated subcategory, then a metric on T T T\mathscr{T}T restricts to a metric on S S S\mathscr{S}S. In particular, if T T T\mathscr{T}T is approximable, the metric on T T T\mathscr{T}T of the previous paragraph restricts to give metrics on the full subcategories T c T c T^(c)\mathscr{T}^{c}Tc and T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}Tcb. Once again these metrics are only defined up to equivalence. And, of course, a metric on S S S\mathscr{S}S is also a metric on S op S op  S^("op ")\mathscr{S}^{\text {op }}Sop , thus we have specified (up to equivalence) canonical metrics on T c , T c b , [ T c ] op T c , T c b , T c op  T^(c),T_(c)^(b),[T^(c)]^("op ")\mathscr{T}^{c}, \mathscr{T}_{c}^{b},\left[\mathscr{T}^{c}\right]^{\text {op }}Tc,Tcb,[Tc]op , and [ T c b ] op T c b op  [T_(c)^(b)]^("op ")\left[\mathscr{T}_{c}^{b}\right]^{\text {op }}[Tcb]op .
Suppose S S S\mathscr{S}S is a triangulated category with a metric. A Cauchy sequence in S S S\mathscr{S}S is a sequence of morphisms E 1 E 2 E 3 E 1 → E 2 → E 3 → ⋯ E_(1)rarrE_(2)rarrE_(3)rarr cdotsE_{1} \rightarrow E_{2} \rightarrow E_{3} \rightarrow \cdotsE1→E2→E3→⋯ which eventually become arbitrarily short.

embeds S S S\mathscr{S}S into the category Mod- S S S\mathscr{S}S of additive functors S op A S op  → A ℓ S^("op ")rarrAℓ\mathscr{S}^{\text {op }} \rightarrow \mathscr{A} \mathscr{\ell}Sop →Aℓ. In the category Mod- S S S\mathscr{S}S colimits exist, allowing us to define
(1) The category L ( S ) L ( S ) L(S)\mathfrak{L}(\mathscr{S})L(S) is the full subcategory of Mod- S S S\mathscr{S}S, whose objects are the colimits of Yoneda images of Cauchy sequences in S S S\mathscr{S}S;
(2) The full subcategory S ( S ) R ( S ) S ( S ) ⊂ R ( S ) S(S)subR(S)\mathscr{S}(\mathscr{S}) \subset \mathbb{R}(\mathscr{S})S(S)⊂R(S) has for objects those functors F R ( S ) F ∈ R ( S ) ⊂ F inR(S)subF \in \mathbb{R}(\mathscr{S}) \subsetF∈R(S)⊂ Mod- S S S\mathscr{S}S which take sufficiently short morphisms to isomorphisms. In symbols, F L ( S ) F ∈ L ( S ) F inL(S)F \in \mathfrak{L}(\mathscr{S})F∈L(S) belongs to S ( S ) S ( S ) S(S)\mathscr{S}(\mathscr{S})S(S) if there exists an ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 such that
{ Length ( f ) < ε } { F ( f ) is an isomorphism } {  Length  ( f ) < ε } ⇒ { F ( f )  is an isomorphism  } {" Length "(f) < epsi}=>{F(f)" is an isomorphism "}\{\text { Length }(f)<\varepsilon\} \Rightarrow\{F(f) \text { is an isomorphism }\}{ Length (f)<ε}⇒{F(f) is an isomorphism }
(3) The exact triangles in S ( S ) S ( S ) S(S)\mathbb{S}(\mathscr{S})S(S) are the colimits in Mod- S S S\mathscr{S}S of Yoneda images of Cauchy sequences of exact triangles in S S S\mathscr{S}S, where the colimits happen to lie in ( S ) ⊆ ( S ) sube(S)\subseteq(\mathscr{S})⊆(S).
A word of caution about (3): if we are given in S S S\mathscr{S}S a Cauchy sequence of exact triangles, we can form the colimit in Mod- S S S\mathscr{S}S of its Yoneda image. This colimit is guaranteed to lie in R ( S ) R ( S ) R(S)\mathfrak{R}(\mathscr{S})R(S), but will not usually lie in the smaller S ( S ) S ( S ) S(S)\mathscr{S}(\mathscr{S})S(S). If it happens to lie in S ( S ) S ( S ) S(S)\mathbb{S}(\mathscr{S})S(S) then (3) declares it to be an exact triangle in S ( S ) S ( S ) S(S)\mathscr{S}(\mathscr{S})S(S).
And now we are ready for the theorem.
Theorem 6.11. Let S S S\mathscr{S}S be a triangulated category with a metric. Assume the metric is good; this is a technical term, see [30, DEFInITION 10] for the precise formulation. Then
(1) The category ( S ) â„‘ ( S ) â„‘(S)\mathbb{\Im}(\mathscr{S})â„‘(S) of Discussion 6.10(2), with the exact triangles as defined in Discussion 6.10(3), is a triangulated category.
Now let T T T\mathscr{T}T be an approximable triangulated category. In Discussion 6.10 we constructed (up to equivalence) a metric on T T T\mathscr{T}T, and hence on its subcategories T c T c T^(c)\mathscr{T}^{c}Tc and [ T c b ] o p T c b o p [T_(c)^(b)]^(op)\left[\mathscr{T}_{c}^{b}\right]^{\mathrm{op}}[Tcb]op. Those metrics are all good, and the theorem goes on to give natural, exact equivalences of triangulated categories
(2) S ( T c ) T c b S T c ≅ T c b S(T^(c))~=T_(c)^(b)\mathbb{S}\left(\mathscr{T}^{c}\right) \cong \mathscr{T}_{c}^{b}S(Tc)≅Tcb. This equivalence is unconditional.
(3) If the approximable triangulated category T T T\mathscr{T}T happens to be noetherian as in [27, DEFINITION 5.1], then ( [ T c b ] op ) [ T c ] op ℑ T c b op  ≅ T c op  ℑ([T_(c)^(b)]^("op "))~=[T^(c)]^("op ")\Im\left(\left[\mathscr{T}_{c}^{b}\right]^{\text {op }}\right) \cong\left[\mathscr{T}^{c}\right]^{\text {op }}ℑ([Tcb]op )≅[Tc]op .
Remark 6.12. First of all, in Theorem 6.11(3) we assumed that the approximable triangulated T T T\mathscr{T}T is noetherian as in [27, DEFInITIon 5.1]. The only observation we want to make here is that if X X XXX is a noetherian, separated scheme then the approximable triangulated category T = D q c ( X ) T = D q c ( X ) T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)T=Dqc(X) is noetherian. Thus, for noetherian, separated schemes X X XXX, Theorem 6.11 gives exact equivalences of triangulated categories
The research paper [27] contains the proofs of the assertions in Theorem 6.11. The reader can find a skimpy survey in [31, SECTION 9] and a more extensive one in [30]. In [31, HISTORICAL SURVEY 9.1] there is a discussion of precursors of the results.

7. FUTURE DIRECTIONS

New scientific developments are tentative and unpolished; only with the passage of time do they acquire the gloss and elegance of a refined, varnished theory. And there is nothing more difficult to predict than the future. My colleague Neil Trudinger used to joke that my beard makes me look like a biblical prophet-the reader should not be deceived, appearances are notoriously misleading, the abundance of facial hair is not a reliable yardstick for measuring the gift of foresight that marks out a visionary, and I am certifiably not a clairvoyant. All I do in this section is offer a handful of obvious questions that spring to mind. The list is not meant to be exhaustive, and might well be missing major tableaux of the overall picture. It is entirely possible that the future will see this theory flourish in directions orthogonal to those sketched here.
Let us begin with what is freshest in our minds: we have just seen Theorem 6.11, part (1) of which tells us that, given a triangulated category S S S\mathscr{S}S with a good metric, there is a recipe producing another triangulated category S ( S ) S ( S ) S(S)\mathscr{S}(\mathscr{S})S(S), which, as it happens, comes with an induced good metric. We can ask:
Problem 7.1. Can one formulate reasonable sufficient conditions, on the triangulated category S S S\mathscr{S}S and on its good metric, to guarantee that ( S ( S ) o p ) = S o p â„‘ S ( S ) o p = S o p â„‘(S(S)^(op))=S^(op)\Im\left(S(\mathscr{S})^{\mathrm{op}}\right)=\mathscr{S}^{\mathrm{op}}â„‘(S(S)op)=Sop ? Who knows, maybe even necessary and sufficient conditions?
Motivating Example 7.2. Let T T T\mathscr{T}T be an approximable triangulated category and put S = T c S = T c S=T^(c)\mathscr{S}=\mathscr{T}^{c}S=Tc, with the metric of Discussion 6.10. Theorem 6.11(2) computes for us that S ( T c ) T c b S T c ≅ T c b S(T^(c))~=T_(c)^(b)\mathbb{S}\left(\mathscr{T}^{c}\right) \cong \mathscr{T}_{c}^{b}S(Tc)≅Tcb. I ask the reader to believe that the natural, induced metric on ( T c ) ₫ T c ₫(T^(c))₫\left(\mathscr{T}^{c}\right)₫(Tc) agrees with the metric on T c b T T c b ⊂ T T_(c)^(b)subT\mathscr{T}_{c}^{b} \subset \mathscr{T}Tcb⊂T given in Discussion 6.10. Now Theorem 6.11(3) goes on to tell us that, as long as the approximable triangulated category T T T\mathscr{T}T is noetherian, we also have that S ( [ T c b ] op ) [ T c ] op S T c b op  ≅ T c op  S([T_(c)^(b)]^("op "))~=[T^(c)]^("op ")\mathbb{S}\left(\left[\mathscr{T}_{c}^{b}\right]^{\text {op }}\right) \cong\left[\mathscr{T}^{c}\right]^{\text {op }}S([Tcb]op )≅[Tc]op ; as it happens, the induced good metric on S ( [ T c b ] op ) S T c b op  S([T_(c)^(b)]^("op "))\mathbb{S}\left(\left[\mathscr{T}_{c}^{b}\right]^{\text {op }}\right)S([Tcb]op ) also agrees, up to equivalence, with the metric that Discussion 6.10 created on [ T c ] op T c op  [T^(c)]^("op ")\left[\mathscr{T}^{c}\right]^{\text {op }}[Tc]op . Combining these we have many examples of exact equivalences of triangulated categories S ( S ( S ) o p ) S o p S S ( S ) o p ≅ S o p S(S(S)^(op))~=S^(op)\mathbb{S}\left(\mathbb{S}(\mathscr{S})^{\mathrm{op}}\right) \cong \mathscr{S}^{\mathrm{op}}S(S(S)op)≅Sop, which are homeomorphisms with respect to the metrics. Thus Problem 7.1 asks the reader to find the right generalization.
Next one can wonder about the functoriality of the construction. Suppose F : S T F : S → T F:SrarrTF: \mathscr{S} \rightarrow \mathscr{T}F:S→T is a triangulated functor, and that both S S S\mathscr{S}S and T T T\mathscr{T}T have good metrics. What are reasonable sufficient conditions which guarantee the existence of an induced functor Ξ ( F ) Ξ ( F ) Xi(F)\Xi(F)Ξ(F), either from S ( S ) S ( S ) S(S)\mathscr{S}(\mathscr{S})S(S) to S ( T ) S ( T ) S(T)\mathscr{S}(\mathscr{T})S(T) or in the other direction? So far there is one known result of this genre, the reader can find the statement below in Sun and Zhang [37, THEOREM 1.1(3)].
Theorem 7.3. Suppose we are given two triangulated categories S S S\mathscr{S}S and T T T\mathscr{T}T, both with good metrics. Suppose we are also given a pair of functors F : S T : G F : S ⇄ T : G F:S⇄T:GF: \mathscr{S} \rightleftarrows \mathscr{T}: GF:S⇄T:G with F G F ⊣ G F⊣GF \dashv GF⊣G, meaning that F F FFF is left adjoint to G G GGG. Assume further that both F F FFF and G G GGG are continuous with respect to the metrics, in the obvious sense.
Then the functor F ^ : Mod T Mod S F ^ : Mod − T → Mod − S hat(F):Mod-Trarr Mod-S\hat{F}: \operatorname{Mod}-\mathscr{T} \rightarrow \operatorname{Mod}-\mathscr{S}F^:Mod−T→Mod−S induced by composition with F F FFF, that is,
the functor taking the T T T\mathscr{T}T-module H : T o p A b H : T o p → A b H:T^(op)rarrAbH: \mathscr{T}^{\mathrm{op}} \rightarrow \mathscr{A} bH:Top→Ab to the S S S\mathscr{S}S-module ( H F ) : S o p A b ( H ∘ F ) : S o p → A b (H@F):S^(op)rarrAb(H \circ F): \mathscr{S}^{\mathrm{op}} \rightarrow \mathscr{A} b(H∘F):Sop→Ab, restricts to a functor which we will denote Ξ ( F ) :⊆ ( T ) ( S ) Ξ ( F ) :⊆ ( T ) → â„‘ ( S ) Xi(F):⊆(T)rarrâ„‘(S)\Xi(F): \subseteq(\mathscr{T}) \rightarrow \mathbb{\Im}(\mathscr{S})Ξ(F):⊆(T)→ℑ(S). That is, the functor Ξ ( F ) Ξ ( F ) Xi(F)\Xi(F)Ξ(F) is defined to be the unique map making the square below commute
where the vertical inclusions are given by the definition of S ( ? ) R ( ? ) S ( ? ) ⊂ R ( ? ) ⊂ S(?)subR(?)sub\mathbb{S}(?) \subset \mathbb{R}(?) \subsetS(?)⊂R(?)⊂ Mod-? of Discus sion 6.10 (1) and (2).
Furthermore, the functor ( F ) ⊆ ( F ) sube(F)\subseteq(F)⊆(F) respects the exact triangles as defined in Discussion 6.10 ( 3 ) 6.10 ( 3 ) 6.10(3)6.10(3)6.10(3).
Sun and Zhang go on to study recollements. Suppose we are given a recollement of triangulated categories
If all three triangulated categories come with good metrics, and if all six functors are continuous, then the following may be found in [37, THEOREM 1.2].
Theorem 7.4. Under the hypotheses above, applying S S S\mathbb{S}S yields a right recollement
In the presence of enough continuous adjoints, we deduce that a semiorthogonal decomposition of S S S\mathscr{S}S gives rise to a semiorthogonal decomposition of S ( S ) S ( S ) S(S)\mathscr{S}(\mathscr{S})S(S). In view of the fact that there are metrics on D perf ( X ) D perf  ( X ) D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)Dperf (X) and D coh b ( X ) D coh  b ( X ) D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X) such that
it is natural to wonder how the recent theorem of Sun and Zhang [37, THEOREM 1.2] compares with the older work of Kuznetsov [19, SECTION 2.5] and [20, SECTION 4].
The above shows that, subject to suitable hypotheses, the construction taking S S S\mathscr{S}S to S ( S ) S ( S ) S(S)\mathscr{S}(\mathscr{S})S(S) can preserve (some of) the internal structure on the category S S S\mathscr{S}S-for example, semiorthogonal decompositions. This leads naturally to
Problem 7.5. What other pieces of the internal structure of S S S\mathscr{S}S are respected by the construction that passes to ( S ) ⊆ ( S ) sube(S)\subseteq(\mathscr{S})⊆(S) ? Under what conditions are these preserved?
Problem 7.5 may sound vague, but it can be made precise enough. For example, there is a huge literature dealing with the group of autoequivalences of the derived categories D coh b ( X ) D coh  b ( X ) D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X). Now, as it happens, the metrics for which Remark 6.12 gives the equivalences
S ( D perf ( X ) ) D coh b ( X ) , ( D c o h b ( X ) o p ) D perf ( X ) o p S D perf  ( X ) ≅ D coh  b ( X ) , ⋐ D c o h b ( X ) o p ≅ D perf  ( X ) o p S(D^("perf ")(X))~=D_("coh ")^(b)(X),quad⋐(D_(coh)^(b)(X)^(op))~=D^("perf ")(X)^(op)\mathbb{S}\left(\mathbf{D}^{\text {perf }}(X)\right) \cong \mathbf{D}_{\text {coh }}^{b}(X), \quad \Subset\left(\mathbf{D}_{\mathrm{coh}}^{b}(X)^{\mathrm{op}}\right) \cong \mathbf{D}^{\text {perf }}(X)^{\mathrm{op}}S(Dperf (X))≅Dcoh b(X),⋐(Dcohb(X)op)≅Dperf (X)op
can be given (up to equivalence) intrinsic descriptions. Note that the way we introduced these metrics, in Discussion 6.10, was to use a preferred t-structure on T = D q c ( X ) T = D q c ( X ) T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)T=Dqc(X) to give on T T T\mathscr{T}T a metric, unique up to equivalence, and hence induced metrics on T c = D perf ( X ) T c = D perf  ( X ) T^(c)=D^("perf ")(X)\mathscr{T}^{c}=\mathbf{D}^{\text {perf }}(X)Tc=Dperf (X) and on T c b = D coh b ( X ) T c b = D coh  b ( X ) T_(c)^(b)=D_("coh ")^(b)(X)\mathscr{T}_{c}^{b}=\mathbf{D}_{\text {coh }}^{b}(X)Tcb=Dcoh b(X) which are also unique up to equivalence. But this description seems to depend on an embedding into the large category T T T\mathscr{T}T. What I am asserting now is that there are alternative descriptions of the same equivalence classes of metrics on T c T c T^(c)\mathscr{T}^{c}Tc and on T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}Tcb, which do not use the embedding into T T T\mathscr{T}T. The interested reader can find this in the later sections of [27]. Anyway, a consequence is that any autoequivalence, of either D perf ( X ) D perf  ( X ) D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)Dperf (X) or of D coh b ( X ) D coh  b ( X ) D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X), must be continuous with a continuous inverse. Hence the group of autoequivalences of D coh b ( X ) D coh  b ( X ) D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X) must be isomorphic to the group of autoequivalences of D perf ( X ) D perf  ( X ) D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)Dperf (X). Or more generally, assume T T T\mathscr{T}T is a noetherian, approximable triangulated category, where noetherian has the meaning of [27, DEFINITION 5.1]. Then the group of exact autoequivalences of T c T c T^(c)\mathscr{T}^{c}Tc is canonically isomorphic to the group of exact autoequivalences of T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}Tcb.
Are there similar theorems about t-structures in S S S\mathscr{S}S going to t-structures in S ( S ) S ( S ) S(S)\mathscr{S}(\mathscr{S})S(S) ? Or about stability conditions on S S S\mathscr{S}S mapping to stability conditions on S ( S ) S ( S ) S(S)\mathscr{S}(\mathscr{S})S(S) ?
We should note that any such theorem will have to come with conditions. After all, the category D coh b ( X ) D coh  b ( X ) D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X) always has a bounded t-structure, while Antieau, Gepner, and Heller [3,
THEOREM 1.1] show that D perf ( X ) D perf  ( X ) D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)Dperf (X) does not in general. Thus it is possible for S S S\mathscr{S}S to have a bounded t-structure but for S ( S ) S ( S ) S(S)\mathscr{S}(\mathscr{S})S(S) not to. And in this particular example, the equivalence class of the metric has an intrinsic description, in the sense mentioned above.
Perhaps we should remind the reader that the article [3], by Antieau, Gepner, and Heller, finds a K K KKK-theoretic obstruction to the existence of bounded t-structures, more precisely if an appropriate category E E E\mathscr{E}E has a bounded t-structure then K 1 ( E ) = 0 K − 1 ( E ) = 0 K_(-1)(E)=0K_{-1}(\mathscr{E})=0K−1(E)=0. Hence the reference to [3] immediately raises the question of how the construction passing from S S S\mathscr{S}S to S ( S ) S ( S ) S(S)\mathbb{S}(\mathscr{S})S(S) might relate to K K KKK-theory, especially to negative K K KKK-theory. Of course, one has to be a little circumspect here. While there is a K K KKK-theory for triangulated categories (see [25] for a survey), this K K KKK-theory has only been proved to behave well for "nice" triangulated categories, for example, for triangulated categories with bounded t-structures. Invariants like negative K K KKK-theory have never been defined for triangulated categories, and might well give nonsense. In what follows we will assume that all the K K KKK-theoretic statements are for triangulated categories with chosen enhancements, and that K K KKK-theory means the Waldhausen K K KKK-theory of the enhancement. We recall in passing that the enhancements are unique for many interesting classes of triangulated categories, see Lunts and Orlov [21], Canonaco and Stellari [9], Antieau [2] and Canonaco, Neeman, and Stellari [8].
With the disclaimers out of the way, what do the results surveyed in this article have to do with negative K K KKK-theory?
Let us begin with Schlichting's conjecture [36, CONJEcTURE 1 OF SECTION 10]; this conjecture, now known to be false [32], predicted that the negative K K KKK-theory of any abelian category should vanish. But Schlichting also proved that (1) K 1 ( A ) = 0 K − 1 ( A ) = 0 K_(-1)(A)=0K_{-1}(\mathscr{A})=0K−1(A)=0 for any abelian category A A A\mathscr{A}A, and (2) K n ( A ) = 0 K − n ( A ) = 0 K_(-n)(A)=0K_{-n}(\mathscr{A})=0K−n(A)=0 whenever A A A\mathscr{A}A is a noetherian abelian category and n > 0 n > 0 n > 0n>0n>0. Now note that the K ( A ) = K ( A o p ) K ( A ) = K A o p K(A)=K(A^(op))K(\mathscr{A})=K\left(\mathscr{A}^{\mathrm{op}}\right)K(A)=K(Aop), hence the negative K K KKK-theory of any artinian abelian category must also vanish. And playing with extensions of abelian categories, we easily deduce the vanishing of the negative K K KKK-theory of a sizeable class of abelian categories. So while Schlichting's conjecture is false in the generality in which it was stated, there is some large class of abelian categories for which it is true. The challenge is to understand this class.
It becomes interesting to see what relation, if any, the results surveyed here have with this question.
Let us begin with Theorems 4.4 and 4.2. Theorem 4.4 tells us that, when X X XXX is a
quasiexcellent, finite-dimensional, separated noetherian scheme, the category D coh b ( X ) D coh  b ( X ) D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X) is strongly generated. This category has a unique enhancement whose K K KKK-theory agrees with the K K KKK-theory of the noetherian abelian category Coh ( X ) Coh ⁡ ( X ) Coh(X)\operatorname{Coh}(X)Coh⁡(X), hence the negative K K KKK-theory vanishes. Theorem 4.2 and Remark 4.3 tell us that the category D perf ( X ) D perf  ( X ) D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)Dperf (X) has a strong generator if and only if X X XXX is regular and finite-dimensional-in which case it is equivalent to D coh b ( X ) D coh  b ( X ) D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X) and its unique enhancement has vanishing negative K K KKK-theory. This raises the question:
Problem 7.6. If T T T\mathscr{T}T is a triangulated category with a strong generator, does it follow that any enhancement of T T T\mathscr{T}T has vanishing negative K K KKK-theory?
Let us refine this question a little. In Definition 4.1 we learned that a strong generator, for a triangulated category T T T\mathscr{T}T, is an object G T G ∈ T G inTG \in \mathscr{T}G∈T such that there exists an integer > 0 ℓ > 0 ℓ > 0\ell>0ℓ>0
with T = G T = ⟨ G ⟩ ℓ T=(:G:)_(ℓ)\mathscr{T}=\langle G\rangle_{\ell}T=⟨G⟩ℓ. Following Rouquier, we can ask for estimates on the integer ℓ ℓ\ellℓ. This leads us to:
Definition 7.7. Let T T T\mathscr{T}T be a triangulated category. The Rouquier dimension of T T T\mathscr{T}T is the smallest integer 0 ℓ ≥ 0 ℓ >= 0\ell \geq 0ℓ≥0 (we allow the possibility = ℓ = ∞ ℓ=oo\ell=\inftyℓ=∞ ), for which there exists an object G G GGG with T = G + 1 T = ⟨ G ⟩ ℓ + 1 T=(:G:)_(ℓ+1)\mathscr{T}=\langle G\rangle_{\ell+1}T=⟨G⟩ℓ+1. See Rouquier [35] for much more about this fascinating invariant.
There is a rich and beautiful literature estimating this invariant and its various cousins-see Rouquier [35] for the origins of the theory, and a host of other places for subsequent developments. For this survey we note only that, for D coh b ( X ) D coh  b ( X ) D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X), the Rouquier dimension is conjectured to be equal to the Krull dimension of X X XXX. But by a conjecture of Weibel [39], now a theorem of Kerz, Strunk, and Tamme [18], the Krull dimension of X X XXX also has a K K KKK theoretic description: the groups K n K n K_(n)K_{n}Kn of the unique enhancement of D perf ( X ) D perf  ( X ) D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)Dperf (X) vanish for all n < dim ( X ) n < − dim ⁡ ( X ) n < -dim(X)n<-\operatorname{dim}(X)n<−dim⁡(X). Recalling that S = D coh b ( X ) S = D coh  b ( X ) S=D_("coh ")^(b)(X)\mathscr{S}=\mathbf{D}_{\text {coh }}^{b}(X)S=Dcoh b(X) is related to D perf ( X ) D perf  ( X ) D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)Dperf (X) by the fact that the construction S S S\mathbb{S}S interchanges them (up to passing to opposite categories, which has no effect on K K KKK-theory), this leads us to ask:
Problem 7.8. Let S S S\mathscr{S}S be a regular (= strongly generated) triangulated category as in Definition 4.1, and let N < N < ∞ N < ooN<\inftyN<∞ be its Rouquier dimension. Is it true that K n K n K_(n)K_{n}Kn vanishes on any enhancement of ( S ) ⋐ ( S ) ⋐(S)\Subset(\mathscr{S})⋐(S), for any metric on S S S\mathscr{S}S and whenever n < N n < − N n < -Nn<-Nn<−N ?
In an entirely different direction, we know that the construction S S S\mathbb{S}S interchanges D perf ( X ) D perf  ( X ) D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)Dperf (X) and D coh b ( X ) D coh  b ( X ) D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X), and that these categories coincide if and only if X X XXX is regular. This leads us to ask:
Problem 7.9. Is there a way to measure the "distance" between S S S\mathscr{S}S and S ( S ) S ( S ) S(S)\mathscr{S}(\mathscr{S})S(S), in such a way that resolution of singularities can be viewed as a process reducing this distance? Who knows, maybe there is even a good metric on S = D perf ( X ) S = D perf  ( X ) S=D^("perf ")(X)\mathscr{S}=\mathbf{D}^{\text {perf }}(X)S=Dperf (X) and/or on S = D coh b ( X ) S ′ = D coh  b ( X ) S^(')=D_("coh ")^(b)(X)\mathscr{S}^{\prime}=\mathbf{D}_{\text {coh }}^{b}(X)S′=Dcoh b(X) such that the construction ₫ ₫₫₫ takes either S S S\mathscr{S}S or S S ′ S^(')\mathscr{S}^{\prime}S′ to an ( S ) ⊆ ( S ) sube(S)\subseteq(\mathscr{S})⊆(S) or ( S ) ₫ S ′ ₫(S^('))₫\left(\mathscr{S}^{\prime}\right)₫(S′) which is D perf ( Y ) = D coh b ( Y ) D perf  ( Y ) = D coh  b ( Y ) D^("perf ")(Y)=D_("coh ")^(b)(Y)\mathbf{D}^{\text {perf }}(Y)=\mathbf{D}_{\text {coh }}^{b}(Y)Dperf (Y)=Dcoh b(Y) for some resolution of singularities Y Y YYY of X X XXX.
While on the subject of regularity ( = = === strong generation):
Problem 7.10. Is there some way to understand which are the approximable triangulated categories T T T\mathscr{T}T for which T c T c T^(c)\mathscr{T}^{c}Tc and/or T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}Tcb are regular?
Theorems 4.2 and 4.4 deal with the case T = D qc ( X ) T = D qc  ( X ) T=D_("qc ")(X)\mathscr{T}=\mathbf{D}_{\text {qc }}(X)T=Dqc (X). Approximability is used in the proofs given in [33] and [4], but only to ultimately reduce to the case of T c = D perf ( X ) T c = D perf  ( X ) T^(c)=D^("perf ")(X)\mathscr{T}^{c}=\mathbf{D}^{\text {perf }}(X)Tc=Dperf (X) with X X XXX an affine scheme-this case turns out to be classical, it was settled already in Kelly's 1965 article [17]. And the diverse precursors of Theorems 4.2 and 4.4, which we have hardly mentioned in the current survey, are also relatively narrow in scope. But presumably there are other proofs out there, yet to be discovered. And new approaches might well lead to generalizations that hold for triangulated categories having nothing to do with algebraic geometry.
Next let us revisit Theorem 6.6, the theorem identifying each of [ T c ] op T c op  [T^(c)]^("op ")\left[\mathscr{T}^{c}\right]^{\text {op }}[Tc]op  (respectively T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}Tcb ) as the finite homological functors on the other. In view of the motivating application, discussed in Remark 6.9, and of the generality of Theorem 6.6, it is natural to wonder:
Problem 7.11. Do GAGA-type theorems have interesting generalizations to other approximable triangulated categories? The reader is invited to check [29, SECTION 8 AND APPENDIX A]: except for the couple of paragraphs in [29, EXAMPLE A.2] everything is formulated in gorgeous generality and might be applicable in other contexts.
In the context of D coh b ( X ) D coh  b ( X ) D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X), where X X XXX is a scheme proper over a noetherian ring R R RRR, there was a wealth of different-looking GAGA-statements before Jack Hall's lovely paper [12] unified them into one. In other words, the category D coh b ( X ) = T c b D coh  b ( X ) = T c b D_("coh ")^(b)(X)=T_(c)^(b)\mathbf{D}_{\text {coh }}^{b}(X)=\mathscr{T}_{c}^{b}Dcoh b(X)=Tcb had many differentlooking incarnations, and it was not until Hall's paper that it was understood that there was one underlying reason why they all coincided.
Hence Problem 7.11 asks whether this pattern is present for other T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}Tcb, in other words for T c b T T c b ⊂ T T_(c)^(b)subT\mathscr{T}_{c}^{b} \subset \mathscr{T}Tcb⊂T where T T T\mathscr{T}T are some other R R RRR-linear, approximable triangulated categories.
And finally:
Problem 7.12. Is there a version of Theorem 6.6 that holds for non-noetherian rings?
There is evidence that something might be true, see Ben-Zvi, Nadler, and Preygel [5, SECTION 3]. But the author has no idea what the right statement ought to be, let alone how to go about proving it.

ACKNOWLEDGMENTS

We would like to thank Asilata Bapat, Jim Borger, Anand Deopurkar, Jack Hall, Bernhard Keller, Tony Licata, and Bregje Pauwels for corrections and improvements to earlier drafts. Needless to say, the flaws that remain are entirely the author's fault.

FUNDING

This work was partially supported by the Australian Research Council, more specifically by grants number DP150102313 and DP200102537.

REFERENCES

[1] L. Alonso Tarrío, A. Jeremías López, and M. J. Souto Salorio, Construction of t t ttt-structures and equivalences of derived categories. Trans. Amer. Math. Soc. 355 (2003), no. 6, 2523-2543 (electronic).
[2] B. Antieau, On the uniqueness of infinity-categorical enhancements of triangulated categories. 2018, arXiv:1812.01526.
[3] B. Antieau, D. Gepner, and J. Heller, K K KKK-theoretic obstructions to bounded t t ttt structures. Invent. Math. 216 (2019), no. 1, 241-300.
[4] K. Aoki, Quasiexcellence implies strong generation. J. Reine Angew. Math. (published online 14 August 2021, 6 pages), see also 2020, arXiv:2009.02013.
[5] D. Ben-Zvi, D. Nadler, and A. Preygel, Integral transforms for coherent sheaves. J. Eur. Math. Soc. (JEMS) 19 (2017), no. 12, 3763-3812.
[6] A. I. Bondal and M. Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3 (2003), no. 1, 1 36 , 258 1 − 36 , 258 1-36,2581-36,2581−36,258.
[7] J. Burke, A. Neeman, and B. Pauwels, Gluing approximable triangulated categories. 2018, arXiv:1806.05342.
[8] A. Canonaco, A. Neeman, and P. Stellari, Uniqueness of enhancements for derived and geometric categories. 2021, arXiv:2101.04404.
[9] A. Canonaco and P. Stellari, Uniqueness of dg enhancements for the derived category of a Grothendieck category. J. Eur. Math. Soc. (JEMS) 20 (2018), no. 11, 2607-2641.
[10] P. J. Freyd, Homotopy is not concrete. In The Steenrod algebra and its applications, pp. 25-34, Lecture Notes in Math. 168, Springer, 1970.
[11] P. J. Freyd, Homotopy is not concrete. Repr. Theory Appl. Categ. 6 (2004), 1-10 (electronic).
[12] J. Hall, GAGA theorems. 2018, arXiv:1804.01976.
[13] L. Illusie, Conditions de finitude relatives. In Théorie des intersections et théorème de Riemann-Roch, pp. 222-273, Lecture Notes in Math. 225, Springer, Berlin, 1971 .
[14] L. Illusie, Existence de résolutions globales. In Théorie des intersections et théorème de Riemann-Roch, pp. 160-221, Lecture Notes in Math. 225, Springer, Berlin, 1971.
[15] L. Illusie, Généralités sur les conditions de finitude dans les catégories dérivées. In Théorie des intersections et théorème de Riemann-Roch, pp. 78-159, Lecture Notes in Math. 225, Springer, Berlin, 1971.
[16] B. Keller and D. Vossieck, Aisles in derived categories. Bull. Soc. Math. Belg. Sér. A 40 (1988), no. 2, 239-253.
[17] G. M. Kelly, Chain maps inducing zero homology maps. Proc. Camb. Philos. Soc. 61 (1965), 847-854.
[18] M. Kerz, F. Strunk, and G. Tamme, Algebraic K K KKK-theory and descent for blow-ups. Invent. Math. 211 (2018), no. 2, 523-577.
[19] A. G. Kuznetsov, Lefschetz decompositions and categorical resolutions of singularities. Selecta Math. (N.S.) 13 (2008), no. 4, 661-696.
[20] A. G. Kuznetsov, Base change for semiorthogonal decompositions. Compos. Math. 147 (2011), no. 3, 852-876.
[21] V. A. Lunts and D. O. Orlov, Uniqueness of enhancement for triangulated categories. J. Amer. Math. Soc. 23 (2010), no. 3, 853-908.
[22] N. Minami, From Ohkawa to strong generation via approximable triangulated categories-a variation on the theme of Amnon Neeman's Nagoya lecture series. In Bousfield classes and Ohkawa's theorem, pp. 17-88, Springer Proc. Math. Stat. 309, Springer, Singapore, 2020.
[23] A. Neeman, The connection between the K K KKK-theory localisation theorem of Thomason, Trobaugh and Yao, and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. Éc. Norm. Supér. 25 (1992), 547-566.
[24] A. Neeman, The Grothendieck duality theorem via Bousfield's techniques and Brown representability. J. Amer. Math. Soc. 9 (1996), 205-236.
[25] A. Neeman, The K K KKK-theory of triangulated categories. In Handbook of K K KKK-theory, vol. 1, 2, pp. 1011-1078, Springer, Berlin, 2005.
[26] A. Neeman, Strong generation of some derived categories of schemes. In Research perspectives CRM Barcelona, pp. 121-127, Trends Math. 5, Springer, 2016.
[27] A. Neeman, The categories T c T c T^(c)\mathscr{T}^{c}Tc and T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}Tcb determine each other. 2018, arXiv:1806.06471.
[28] A. Neeman, The category [ T c ] op T c op  [T^(c)]^("op ")\left[\mathscr{T}^{c}\right]^{\text {op }}[Tc]op  as functors on T c b T c b T_(c)^(b)\mathscr{T}_{c}^{b}Tcb. 2018, arXiv:1806.05777.
[29] A. Neeman, Triangulated categories with a single compact generator and a Brown representability theorem. 2018, arXiv:1804.02240.
[30] A. Neeman, Metrics on triangulated categories. J. Pure Appl. Algebra 224 (2020), no. 4 , 106206 , 13 4 , 106206 , 13 4,106206,134,106206,134,106206,13.
[31] A. Neeman, Approximable triangulated categories. In Representations of algebras, geometry and physics, pp. 111-155, Contemp. Math. 769, Amer. Math. Soc., Providence, RI, 2021.
[32] A. Neeman, A counterexample to vanishing conjectures for negative K K KKK-theory. Invent. Math. 225 (2021), no. 2, 427-452.
[33] A. Neeman, Strong generators in D perf ( X ) D perf  ( X ) D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)Dperf (X) and D coh b ( X ) D coh  b ( X ) D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X). Ann. of Math. (2) 193 (2021), no. 3, 689-732.
[34] D. O. Orlov, Smooth and proper noncommutative schemes and gluing of DG categories. Adv. Math. 302 (2016), 59-105.
[35] R. Rouquier, Dimensions of triangulated categories. J. K-Theory 1 (2008), no. 2, 193-256.
[36] M. Schlichting, Negative K K KKK-theory of derived categories. Math. Z. 253 (2006), no. 1 , 97 134 1 , 97 − 134 1,97-1341,97-1341,97−134.
[37] Y. Sun and Y. Zhang, Ladders and completion of triangulated categories. Theory Appl. Categ. 37 (2021), Paper No. 4, 95-106.
[38] J.-L. Verdier, Des catégories dérivées des catégories abéliennes. Astérisque (1996), no. 239, xii+253 pp. (1997). With a preface by Luc Illusie, edited and with a note by Georges Maltsiniotis.
[39] C. A. Weibel, K-theory and analytic isomorphisms. Invent. Math. 61 (1980), no. 2, 177-197.

AMNON NEEMAN

Mathematical Sciences Institute, Building 145, The Australian National University, Canberra, ACT 2601, Australia, Amnon.Neeman@ anu.edu.au

SYZYGIES OVER A POLYNOMIAL RING

IRENA PEEVA

ABSTRACT

We discuss results and open problems on graded minimal free resolutions over polynomial rings.

MATHEMATICS SUBJECT CLASSIFICATION 2020

13D02

KEYWORDS

Syzygies, free resolutions, Betti numbers

1. INTRODUCTION

Research on free resolutions is a core and beautiful area in Commutative Algebra. It contains a number of challenging conjectures and open problems; some of them are discussed in the book [101].
For simplicity, we will work throughout over the polynomial ring S = C [ x 1 , , x n ] S = C x 1 , … , x n S=C[x_(1),dots,x_(n)]S=\mathbb{C}\left[x_{1}, \ldots, x_{n}\right]S=C[x1,…,xn], which is standard graded by deg ( x i ) = 1 deg ⁡ x i = 1 deg(x_(i))=1\operatorname{deg}\left(x_{i}\right)=1deg⁡(xi)=1 for every i i iii. Many of the results work in much bigger generality; for example, over any field, or over some graded quotient rings of S S SSS. We leave it to the interested reader to look for the precise generality of the results using the references. We focus on some main ideas about finite resolutions which are present over polynomial rings.
The idea to describe the structure of a module by a free resolution was introduced by Hilbert in his famous paper [76]; this approach was present in the work of Cayley [35] as well. Every finitely generated S S SSS-module T T TTT has a free resolution. If T T TTT is graded, there exists a minimal free resolution F T F T F_(T)\mathbf{F}_{T}FT which is unique up to an isomorphism and is contained in any free resolution of T T TTT. Hilbert's insight was that the properties of the minimal free resolution F T F T F_(T)\mathbf{F}_{T}FT are closely related to the invariants of the resolved module T T TTT. The key point is that the resolution can be interpreted as an exact complex of finitely generated free modules F i F i F_(i)F_{i}Fi so that
Thus, the resolution is a way of describing the structure of T T TTT.
The condition of minimality is important. The mere existence of free resolutions suffices for computing Hilbert functions and for foundational issues such as the definition of Ext and Tor. However, without minimality, resolutions are not unique, and the uniformity of constructions of nonminimal resolutions (like the Bar resolution) implies that they give little insight into the structure of the resolved modules. In contrast, the minimal free resolution F T F T F_(T)\mathbf{F}_{T}FT encodes a lot of properties of T T TTT; for example, the Auslander-Buchsbaum formula expresses the depth of T T TTT in terms of the length (called projective dimension) of F T F T F_(T)\mathbf{F}_{T}FT, while nonminimal resolutions do not measure depth.
Free resolutions have applications in mathematical fields as diverse as Algebraic Geometry, Combinatorics, Computational Algebra, Invariant Theory, Mathematical Physics, Noncommutative Algebra, Number Theory, and Subspace Arrangements. For many years, they have been both central objects and fruitful tools in Commutative Algebra.
The connections of resolutions to Algebraic Geometry are especially rich, and the book [51] is focussed on that. One of the most challenging open problems in this area, which remains open to this date, is Green's conjecture; see the recent paper by AproduFarkas-Papadima-Raicu-Weyman [5] for more details on this problem.
It should be noted that the world of minimal free resolutions is much wider and diverse than graded resolutions over polynomial rings. Resolutions are studied in other major situations, and there are many important and exciting results and open problems there. For example, there is an extensive research in the multigraded case, which contains resolutions of monomial ideals, resolutions of toric ideals, and resolutions of binomial edge ideals. Another fascinating and important area is the study of minimal free resolutions over quotient rings; such resolutions are usually infinite (by a theorem of Serre) and so their properties are quite different than what we see in finite resolutions over a polynomial ring. An interesting new idea is the recent introduction of virtual resolutions by Berkesch-Erman-Smith [13].

2. FREE RESOLUTIONS

A free resolution of a finitely generated S S SSS-module T T TTT is an exact sequence
F : F 2 d 2 F 1 d 1 F 0 d 0 T 0 F : … → F 2 → d 2 F 1 → d 1 F 0 → d 0 T → 0 F:quad dots rarrF_(2)rarr"d_(2)"F_(1)rarr"d_(1)"F_(0)rarr"d_(0)"T rarr0\mathbf{F}: \quad \ldots \rightarrow F_{2} \xrightarrow{d_{2}} F_{1} \xrightarrow{d_{1}} F_{0} \xrightarrow{d_{0}} T \rightarrow 0F:…→F2→d2F1→d1F0→d0T→0
of homomorphisms of free finitely generated S S SSS-modules F i F i F_(i)F_{i}Fi. The maps d i d i d_(i)d_{i}di are called differentials.
If T T TTT is graded, there exists a minimal free resolution F T F T F_(T)\mathbf{F}_{T}FT which is unique up to an isomorphism and is contained in any free resolution of T T TTT (see [101, THEOREM 7.5], [101, THEOREM 3.5]). Minimality can be characterized in the following simple way: F F F\mathbf{F}F is minimal if
d i + 1 ( F i + 1 ) ( x 1 , , x n ) F i for all i 0 d i + 1 F i + 1 ⊆ x 1 , … , x n F i  for all  i ≥ 0 d_(i+1)(F_(i+1))sube(x_(1),dots,x_(n))F_(i)quad" for all "i >= 0d_{i+1}\left(F_{i+1}\right) \subseteq\left(x_{1}, \ldots, x_{n}\right) F_{i} \quad \text { for all } i \geq 0di+1(Fi+1)⊆(x1,…,xn)Fi for all i≥0
that is, no invertible elements appear in the differential matrices.
Hilbert's intuition was that the properties of the minimal free resolution F T F T F_(T)\mathbf{F}_{T}FT are closely related to the invariants of the resolved module T T TTT. The key point is that the map d 0 : F 0 T d 0 : F 0 → T d_(0):F_(0)rarr Td_{0}: F_{0} \rightarrow Td0:F0→T sends a basis of F 0 F 0 F_(0)F_{0}F0 to a minimal system E E E\mathcal{E}E of generators of T T TTT, the first differential d 1 d 1 d_(1)d_{1}d1 describes the minimal relations R R R\mathcal{R}R among the generators E E E\mathscr{E}E, the second differential d 2 d 2 d_(2)d_{2}d2 describes the minimal relations on the relations R R R\mathcal{R}R, etc.; see (1.1). Hilbert's Syzygy Theorem 4.1 is a fundamental result on the structure of such resolutions and leads to many applications. It shows that every finitely generated graded S S SSS-module has a finite free resolution (that is, F j = 0 F j = 0 F_(j)=0F_{j}=0Fj=0 for j 0 j ≫ 0 j≫0j \gg 0j≫0 ).
The submodule Im ( d i ) = Ker ( d i 1 ) Im ⁡ d i = Ker ⁡ d i − 1 Im(d_(i))=Ker(d_(i-1))\operatorname{Im}\left(d_{i}\right)=\operatorname{Ker}\left(d_{i-1}\right)Im⁡(di)=Ker⁡(di−1) of F i 1 F i − 1 F_(i-1)F_{i-1}Fi−1 is called the i i iii th syzygy module of T T TTT, and its elements are called i i iii th syzygies.

3. BETTI NUMBERS

Let T T TTT be a graded finitely generated S S SSS-module. The differentials in the minimal free resolution F T F T F_(T)\mathbf{F}_{T}FT of T T TTT are often very intricate, and so it may be more fruitful to focus on numerical invariants. The rank of the free module F i F i F_(i)F_{i}Fi in F T F T F_(T)\mathbf{F}_{T}FT is called the i i iii th Betti number and is denoted by b i ( T ) b i ( T ) b_(i)(T)b_{i}(T)bi(T). It may be expressed as
b i ( T ) = dim Tor i S ( T , C ) = dim Ext S i ( T , C ) b i ( T ) = dim ⁡ Tor i S ⁡ ( T , C ) = dim ⁡ Ext S i ⁡ ( T , C ) b_(i)(T)=dim Tor_(i)^(S)(T,C)=dim Ext_(S)^(i)(T,C)b_{i}(T)=\operatorname{dim} \operatorname{Tor}_{i}^{S}(T, \mathbb{C})=\operatorname{dim} \operatorname{Ext}_{S}^{i}(T, \mathbb{C})bi(T)=dim⁡ToriS⁡(T,C)=dim⁡ExtSi⁡(T,C)
The Betti numbers are extensively studied numerical invariants of T T TTT, and they encode a lot of information about the module.
Note that in the graded case we have graded Betti numbers b i , j ( T ) b i , j ( T ) b_(i,j)(T)b_{i, j}(T)bi,j(T) : Since T T TTT is graded, it has a graded minimal free resolution, that is, the differentials preserve degree (they are homogeneous maps of degree 0). Thus, we have graded Betti numbers
b i , j ( T ) = dim Tor i S ( T , C ) j = dimExt S i ( T , C ) j b i , j ( T ) = dim ⁡ Tor i S ⁡ ( T , C ) j = dimExt S i ⁡ ( T , C ) j b_(i,j)(T)=dim Tor_(i)^(S)(T,C)_(j)=dimExt_(S)^(i)(T,C)_(j)b_{i, j}(T)=\operatorname{dim} \operatorname{Tor}_{i}^{S}(T, \mathbb{C})_{j}=\operatorname{dimExt}_{S}^{i}(T, \mathbb{C})_{j}bi,j(T)=dim⁡ToriS⁡(T,C)j=dimExtSi⁡(T,C)j
Hilbert showed how to use them in order to compute the Hilbert series i = 0 t i dim C ( T i ) ∑ i = 0   t i dim C ⁡ T i sum_(i=0)t^(i)dim_(C)(T_(i))\sum_{i=0} t^{i} \operatorname{dim}_{\mathbb{C}}\left(T_{i}\right)∑i=0tidimC⁡(Ti) which measures the size of the module T T TTT; see [101, THEOREM 16.2].
The graded Betti numbers can be assembled in the Betti table β ( T ) β ( T ) beta(T)\beta(T)β(T), which has entry b i , i + j = b i , i + j ( T ) b i , i + j = b i , i + j ( T ) b_(i,i+j)=b_(i,i+j)(T)b_{i, i+j}=b_{i, i+j}(T)bi,i+j=bi,i+j(T) in position i , j i , j i,ji, ji,j. Following the conventions in the computer algebra system Macaulay 2 [68], the columns of β ( T ) β ( T ) beta(T)\beta(T)β(T) are indexed from left to right by homological degree, and the rows are indexed increasingly from top to bottom. For example, if T T TTT is generated in nonnegative degrees then the Betti table β ( T ) β ( T ) beta(T)\beta(T)β(T) has the form:
0 1 2 ⋯ cdots\cdots⋯
0 : 0 : 0:0:0: b 0 , 0 b 0 , 0 b_(0,0)b_{0,0}b0,0 b 1 , 1 b 1 , 1 b_(1,1)b_{1,1}b1,1 b 2 , 2 b 2 , 2 b_(2,2)b_{2,2}b2,2 ⋯ cdots\cdots⋯
1 : 1 : 1:1:1: b 0 , 1 b 0 , 1 b_(0,1)b_{0,1}b0,1 b 1 , 2 b 1 , 2 b_(1,2)b_{1,2}b1,2 b 2 , 3 b 2 , 3 b_(2,3)b_{2,3}b2,3 ⋯ cdots\cdots⋯
2 : 2 : 2:2:2: b 0 , 2 b 0 , 2 b_(0,2)b_{0,2}b0,2 b 1 , 3 b 1 , 3 b_(1,3)b_{1,3}b1,3 b 2 , 4 b 2 , 4 b_(2,4)b_{2,4}b2,4 ⋯ cdots\cdots⋯
â‹® vdots\vdotsâ‹® â‹® vdots\vdotsâ‹® â‹® vdots\vdotsâ‹® â‹® vdots\vdotsâ‹®
0 1 2 cdots 0: b_(0,0) b_(1,1) b_(2,2) cdots 1: b_(0,1) b_(1,2) b_(2,3) cdots 2: b_(0,2) b_(1,3) b_(2,4) cdots vdots vdots vdots vdots | | 0 | 1 | 2 | $\cdots$ | | :--- | :--- | :--- | :--- | :--- | | $0:$ | $b_{0,0}$ | $b_{1,1}$ | $b_{2,2}$ | $\cdots$ | | $1:$ | $b_{0,1}$ | $b_{1,2}$ | $b_{2,3}$ | $\cdots$ | | $2:$ | $b_{0,2}$ | $b_{1,3}$ | $b_{2,4}$ | $\cdots$ | | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | |
The main general open-ended question on Betti numbers is:
Question 3.1. How do the properties of the (graded) Betti numbers relate to the structure of the minimal free resolution of T T TTT and/or the structure of T T TTT ?
The BEH Conjecture is a long-standing open conjecture on Betti numbers:
BEH Conjecture 3.2 (Buchsbaum-Eisenbud, Horrocks, [20,73]). If T is a finitely generated graded artinian S S SSS-module (artinian means that the module has finite length), then
b i ( T ) ( n i ) for i 0 b i ( T ) ≥ ( n i )  for  i ≥ 0 b_(i)(T) >= ((n)/(i))quad" for "i >= 0b_{i}(T) \geq\binom{ n}{i} \quad \text { for } i \geq 0bi(T)≥(ni) for i≥0
Essentially, the conjecture states that the Koszul resolution (see [101, sEction 14]) of the residue field C C C\mathbb{C}C is the smallest minimal free resolution of an artinian module.
If the above conjecture holds, then it easily follows that we get a lower bound on the Betti numbers for any module (not necessarily artinian) in terms of its codimension; see [17]. The expository papers [17] by Boocher-Grifo and [36] by Charalambous-Evans provide nice overviews on the scarce positive results that are known so far; for example, Herzog-Kühl [74] proved the desired inequalities for linear resolutions. The best currently known result is:
Theorem 3.3 (Walker, [108]). If T T TTT is a finitely generated graded artinian S S SSS-module, then
i = 0 n b i ( T ) 2 n ∑ i = 0 n   b i ( T ) ≥ 2 n sum_(i=0)^(n)b_(i)(T) >= 2^(n)\sum_{i=0}^{n} b_{i}(T) \geq 2^{n}∑i=0nbi(T)≥2n
and equality holds if and only if T T TTT is a complete intersection.
People have wondered how sharp the above bound is when the module is not a complete intersection (that is, T T TTT is not a quotient ring by a regular sequence):
Question 3.4 (Charalambous-Evans-Miller, [37]). If T T TTT is a finitely generated graded artinian S S SSS-module that is not a complete intersection, then do we have
i = 0 n b i ( T ) 2 n + 2 n 1 ? ∑ i = 0 n   b i ( T ) ≥ 2 n + 2 n − 1 ? sum_(i=0)^(n)b_(i)(T) >= 2^(n)+2^(n-1)?\sum_{i=0}^{n} b_{i}(T) \geq 2^{n}+2^{n-1} ?∑i=0nbi(T)≥2n+2n−1?
There are many questions that one may ask and study about Betti numbers when restricted to special classes of modules; most ambitiously, we would like to have a characterization of the sequences that are Betti numbers. A recent result of this kind is the Boij-Söderberg theory, which was conjectured by Boij-Söderberg [16], and proved soon after that. Eisenbud-Fløystad-Weyman proved the characteristic-zero case in [52] and then Eisenbud-Schreyer dealt with any characteristic in [56]. Later, efficient methods for such constructions were given by Berkesch, Kummini, Erman, Sam in [14] and by Fløystad in [61, SECTION 3]. The expository papers [60,62] provide nice overviews of this topic.

4. PROJECTIVE DIMENSION

Projective dimension and regularity are the main numerical invariants that measure the complexity of a minimal free resolution. We will discuss regularity in the next section.
The projective dimension of a graded finitely generated S S SSS-module T T TTT is
pd ( T ) = max { i b i ( T ) 0 } pd ⁡ ( T ) = max i ∣ b i ( T ) ≠ 0 pd(T)=max{i∣b_(i)(T)!=0}\operatorname{pd}(T)=\max \left\{i \mid b_{i}(T) \neq 0\right\}pd⁡(T)=max{i∣bi(T)≠0}
and it is the index of the last nonzero column of the Betti table β ( T ) β ( T ) beta(T)\beta(T)β(T), so it measures the width of the table.
Hilbert's Syzygy Theorem 4.1 (see [101, THEOREM 15.2]). The minimal graded free resolution of a finitely generated graded S S SSS-module is finite, and its projective dimension is at most n n nnn (recall that n n nnn is the number of variables in the polynomial ring S S SSS ).
Hilbert's Syzygy Theorem 4.1 provides a nice upper bound on the projective dimension in terms of the number of variables in the polynomial ring. One may wonder if the number of minimal generators of an ideal can be used to get another nice upper bound on projective dimension. The answer turns out to be negative. A construction of Burch [21] and Kohn [79] produces ideals with 3 generators whose projective dimension is arbitrarily large. Later Bruns [18] showed that the minimal free resolutions of three-generated ideals capture all the pathology of minimal free resolutions of modules. However, the degrees of the generators in these constructions are forced to grow large. Motivated by computational complexity issues, Stillman raised the following question:
Question 4.2 (Stillman, [102, PRobLem 3.14]). Fix an m 1 m ≥ 1 m >= 1m \geq 1m≥1 and a sequence of natural numbers a 1 , , a m a 1 , … , a m a_(1),dots,a_(m)a_{1}, \ldots, a_{m}a1,…,am. Is there a number p p ppp such that pd ( I ) p pd ⁡ ( I ) ≤ p pd(I) <= p\operatorname{pd}(I) \leq ppd⁡(I)≤p for every homogeneous ideal I I III with
a minimal system of generators of degrees a 1 , , a m a 1 , … , a m a_(1),dots,a_(m)a_{1}, \ldots, a_{m}a1,…,am in a polynomial ring? Note that the number of variables in the polynomial ring is not fixed.
A positive answer is provided by:
Theorem 4.3 (Ananyan-Hochster, [4]). Stillman's Question 4.2 has a positive answer.
Other proofs were later given by Erman-Sam-Snowden [58] and Draisma-LasońLeykin [50]. Yet, there are many open questions motivated by a desire to get better upper bounds since the known bounds are quite large. See the recent paper by Caviglia-Liang [29] for some explicit bounds.
Families of ideals with large projective dimension were constructed by McCullough in [89] and by Beder, McCullough, Núñez-Betancourt, Seceleanu, Snapp, Stone in [12]. Such constructions indicate that finding tight bounds could be difficult. Many results dealing with special cases are known in this direction. The expository papers [62,94] provide nice overviews of this topic.

5. REGULARITY

Let L L LLL be a homogeneous ideal in S S SSS. The height of the Betti table of L L LLL is measured by the index of the last nonzero row, and is called the (Castelnuovo-Mumford) regularity of L L LLL, so
reg ( L ) = max { j there exists an i such that b i , i + j ( L ) 0 } reg ⁡ ( L ) = max j ∣  there exists an  i  such that  b i , i + j ( L ) ≠ 0 reg(L)=max{j∣" there exists an "i" such that "b_(i,i+j)(L)!=0}\operatorname{reg}(L)=\max \left\{j \mid \text { there exists an } i \text { such that } b_{i, i+j}(L) \neq 0\right\}reg⁡(L)=max{j∣ there exists an i such that bi,i+j(L)≠0}
Note that reg ( L ) < reg ⁡ ( L ) < ∞ reg(L) < oo\operatorname{reg}(L)<\inftyreg⁡(L)<∞ by Hilbert's Syzygy Theorem 4.1. An important role of regularity is that it measures the complexity of the minimal free resolution of L L LLL, in the sense that it shows up to what degree we have nonvanishing Betti numbers. It has several other important roles.
The definition of regularity implies that it provides an upper bound on the generating degree, namely
reg ( L ) maxdeg ( L ) reg ⁡ ( L ) ≥ maxdeg ⁡ ( L ) reg(L) >= maxdeg(L)\operatorname{reg}(L) \geq \operatorname{maxdeg}(L)reg⁡(L)≥maxdeg⁡(L)
where maxdeg ( L ) maxdeg ⁡ ( L ) maxdeg(L)\operatorname{maxdeg}(L)maxdeg⁡(L) is the maximal degree of an element in a minimal system of homogeneous generators of L L LLL.
Another role of regularity is that it identifies how high we have to truncate an ideal in order to get a linear resolution; we say that a graded ideal has an r r rrr-linear resolution if the ideal is generated in degree r r rrr and the entries in the differential maps in its minimal free resolution are linear.
Theorem 5.1 (see [101, THEOREM 19.7]). Let L L LLL be a graded ideal in S S SSS. If r reg ( L ) r ≥ reg ⁡ ( L ) r >= reg(L)r \geq \operatorname{reg}(L)r≥reg⁡(L) then
L r := L ( i r S i ) L ≥ r := L ∩ ⨁ i ≥ r   S i L_( >= r):=L nn(bigoplus_(i >= r)S_(i))L_{\geq r}:=L \cap\left(\bigoplus_{i \geq r} S_{i}\right)L≥r:=L∩(⨁i≥rSi)
has an r r rrr-linear minimal free resolution, equivalently,
reg ( L r ) = r reg ⁡ L ≥ r = r reg(L_( >= r))=r\operatorname{reg}\left(L_{\geq r}\right)=rreg⁡(L≥r)=r
Another role of regularity is related to Gröbner basis computation. Many computer computations in Commutative Algebra and Algebraic Geometry are based on Gröbner basis theory. It is used, for example, in the computer algebra systems Cocoa [1], Macaulay2 [68], Singular [49]. It is proved by Bayer-Stillman [9] that in generic coordinates and with respect to revlex order, one has to compute up to degree reg ( L ) ( L ) (L)(L)(L) in order to compute a Gröbner basis of L L LLL. This means that reg ( L ) ( L ) (L)(L)(L) is the degree-complexity of the Gröbner basis computation.
Yet another role of regularity is that it can be defined in terms of vanishing of local cohomology modules. See the expository paper [19] for a detailed discussion.
The expository papers [ 38 , 39 ] [ 38 , 39 ] [38,39][38,39][38,39] provide nice overviews of the properties of regularity.
In the rest of this section, we discuss bounds on regularity.
The projective dimension pd ( L ) pd ⁡ ( L ) pd(L)\operatorname{pd}(L)pd⁡(L) of L L LLL is bounded above by the number of variables n n nnn in S S SSS by Hilbert's Syzygy Theorem 4.1. This bound is very nice in several ways: it is small, involves only one parameter, and is given by a simple formula. One may hope that similarly, a nice upper bound on regularity exists. In contrast, the upper bound on regularity involving n n nnn is doubly exponential. Bayer-Mumford (see [8, тHEOREM 3.7]) and Caviglia-Sbarra [32] proved:
Theorem 5.2 (Bayer-Mumford [8], Caviglia-Sbarra [32]). Let L be a graded ideal in S S SSS. Then
reg ( L ) ( 2 maxdeg ( L ) ) 2 n 2 reg ⁡ ( L ) ≤ ( 2 maxdeg ⁡ ( L ) ) 2 n − 2 reg(L) <= (2maxdeg(L))^(2^(n-2))\operatorname{reg}(L) \leq(2 \operatorname{maxdeg}(L))^{2^{n-2}}reg⁡(L)≤(2maxdeg⁡(L))2n−2
where maxdeg ( L ) maxdeg ⁡ ( L ) maxdeg(L)\operatorname{maxdeg}(L)maxdeg⁡(L) is the maximal degree of an element in a minimal system of homogeneous generators of L L LLL.
This bound is nearly sharp. The Mayr-Meyer construction [88] leads to examples of families of ideals attaining high regularity. The following three types of families of ideals attaining doubly exponential regularity were constructed by Bayer-Mumford [8], BayerStillman [10], and Koh [78]:
Theorem 5.3. (1) (Bayer-Stillman, [10, THEOREM 2.6]) For r 1 r ≥ 1 r >= 1r \geq 1r≥1, there exists a homogeneous ideal I r I r I_(r)I_{r}Ir (using d = 3 d = 3 d=3d=3d=3 in their notation) in a polynomial ring with 10 r + 11 10 r + 11 10 r+1110 r+1110r+11 variables for which
maxdeg ( I r ) = 5 reg ( I r ) 3 2 r 1 maxdeg ⁡ I r = 5 reg ⁡ I r ≥ 3 2 r − 1 {:[maxdeg(I_(r))=5],[reg(I_(r)) >= 3^(2^(r-1))]:}\begin{aligned} \operatorname{maxdeg}\left(I_{r}\right) & =5 \\ \operatorname{reg}\left(I_{r}\right) & \geq 3^{2^{r-1}} \end{aligned}maxdeg⁡(Ir)=5reg⁡(Ir)≥32r−1
(2) (Bayer-Mumford, [8, PROPOSItIon 3.11]) For r 1 r ≥ 1 r >= 1r \geq 1r≥1, there exists a homogeneous ideal I r I r I_(r)I_{r}Ir in 10 r + 1 10 r + 1 10 r+110 r+110r+1 variables for which
maxdeg ( I r ) = 4 reg ( I r ) 2 2 r maxdeg ⁡ I r = 4 reg ⁡ I r ≥ 2 2 r {:[maxdeg(I_(r))=4],[reg(I_(r)) >= 2^(2^(r))]:}\begin{aligned} \operatorname{maxdeg}\left(I_{r}\right) & =4 \\ \operatorname{reg}\left(I_{r}\right) & \geq 2^{2^{r}} \end{aligned}maxdeg⁡(Ir)=4reg⁡(Ir)≥22r
(3) (Koh, [78]) For r 1 r ≥ 1 r >= 1r \geq 1r≥1, there exists a homogeneous I r I r I_(r)I_{r}Ir generated by 22 r 2 22 r − 2 22 r-222 r-222r−2 quadrics in a polynomial ring with 22 r 22 r 22 r22 r22r variables for which
maxdeg ( I r ) = 2 reg ( I r ) 2 2 r 1 maxdeg ⁡ I r = 2 reg ⁡ I r ≥ 2 2 r − 1 {:[maxdeg(I_(r))=2],[reg(I_(r)) >= 2^(2^(r-1))]:}\begin{aligned} \operatorname{maxdeg}\left(I_{r}\right) & =2 \\ \operatorname{reg}\left(I_{r}\right) & \geq 2^{2^{r-1}} \end{aligned}maxdeg⁡(Ir)=2reg⁡(Ir)≥22r−1
Further examples of ideals with high regularity were produced by Beder et. al. [12], Caviglia [23], Chardin-Fall [41], and Ullery [107].
Despite these examples of high regularity, there are many important and interesting cases where regularity is bounded by (or equal to) a nice formula and is not dramatically large. As always, the following open-ended problem is of high interest:
Problem 5.4. Find important and interesting cases where regularity is bounded by (or equal to) a nice formula and is not dramatically large.

6. REGULARITY OF PRIME IDEALS

Regularity was studied in Algebraic Geometry as well. In that setting, much better bounds than the doubly-exponential bound discussed in Theorem 5.2, are expected for the regularity of the defining ideals of geometrically nice projective varieties. Lazarsfeld's book [86, SECTION 1.8] and the introduction of the paper [84] by Kwak-Park provide nice overviews of that point of view. In fact, the concept of regularity was introduced by Mumford [98] and generalizes ideas of Castelnuovo. The relation between the definitions of regularity of a coherent sheaf and regularity of a graded ideal (or module) is given in Eisenbud-Goto [53], and may be also found in [51, PROPOSITION 4.16].
Consider a nondegenerate projective variety X P n 1 X ⊂ P n − 1 X subP^(n-1)X \subset \mathbb{P}^{n-1}X⊂Pn−1, that is, X X XXX does not lie on a hyperplane in P n 1 P n − 1 P^(n-1)\mathbb{P}^{n-1}Pn−1.
Some nice bounds were proved in the smooth case. The following bound follows from a more general result by Bertram-Ein-Lazarsfeld [15]:
Theorem 6.1 (Bertram-Ein-Lazarsfeld, [15]). Let X P n 1 X ⊂ P n − 1 X subP^(n-1)X \subset \mathbb{P}^{n-1}X⊂Pn−1 be a smooth irreducible projective variety. If X X XXX is cut out scheme-theoretically by hypersurfaces of degree s ≤ s <= s\leq s≤s, then
reg ( X ) 1 + ( s 1 ) codim ( X ) reg ⁡ ( X ) ≤ 1 + ( s − 1 ) codim ⁡ ( X ) reg(X) <= 1+(s-1)codim(X)\operatorname{reg}(X) \leq 1+(s-1) \operatorname{codim}(X)reg⁡(X)≤1+(s−1)codim⁡(X)
This result was generalized in [42] and [48]. See also [38] for an overview.
Theorem 6.2 (Mumford, [8, THEOREM 3.12]). If X P n 1 X ⊂ P n − 1 X subP^(n-1)X \subset \mathbb{P}^{n-1}X⊂Pn−1 is a nondegenerate smooth projective variety, then
reg ( X ) ( dim ( X ) + 1 ) ( deg ( X ) 2 ) + 2 reg ⁡ ( X ) ≤ ( dim ⁡ ( X ) + 1 ) ( deg ⁡ ( X ) − 2 ) + 2 reg(X) <= (dim(X)+1)(deg(X)-2)+2\operatorname{reg}(X) \leq(\operatorname{dim}(X)+1)(\operatorname{deg}(X)-2)+2reg⁡(X)≤(dim⁡(X)+1)(deg⁡(X)−2)+2
This bound was improved by Kwak-Park as follows:
Theorem 6.3 (Kwak-Park, [84, тHEOREM c]). If X P n 1 X ⊂ P n − 1 X subP^(n-1)X \subset \mathbb{P}^{n-1}X⊂Pn−1 is a nondegenerate smooth projective variety with codim ( X ) 2 codim ⁡ ( X ) ≥ 2 codim(X) >= 2\operatorname{codim}(X) \geq 2codim⁡(X)≥2, then
reg ( X ) dim ( X ) ( deg ( X ) 2 ) + 1 reg ⁡ ( X ) ≤ dim ⁡ ( X ) ( deg ⁡ ( X ) − 2 ) + 1 reg(X) <= dim(X)(deg(X)-2)+1\operatorname{reg}(X) \leq \operatorname{dim}(X)(\operatorname{deg}(X)-2)+1reg⁡(X)≤dim⁡(X)(deg⁡(X)−2)+1
In the influential paper [8], Bayer and Mumford wrote:
"...the main missing piece of information between the general case and the geometrically nice smooth case is that we do not have yet a reasonable bound on the regularity of all reduced equidimensional ideals."
Note that the bounds in the above theorems involve two parameters; for example, dim ( X ) dim ⁡ ( X ) dim(X)\operatorname{dim}(X)dim⁡(X) and deg ( X ) deg ⁡ ( X ) deg(X)\operatorname{deg}(X)deg⁡(X) are used in Theorem 6.2. The following bound involving only deg ( X ) deg ⁡ ( X ) deg(X)\operatorname{deg}(X)deg⁡(X) was first considered in the smooth case:
reg ( X ) deg ( X ) reg ⁡ ( X ) ≤ deg ⁡ ( X ) reg(X) <= deg(X)\operatorname{reg}(X) \leq \operatorname{deg}(X)reg⁡(X)≤deg⁡(X)
It was conjectured by Eisenbud-Goto [53] for any reduced and irreducible nondegenerate variety, and they expected that it might even hold for reduced equidimensional X X XXX which are connected in codimension 1 [8]. In fact, they conjectured the more refined bound
reg ( X ) deg ( X ) codim ( X ) + 1 reg ⁡ ( X ) ≤ deg ⁡ ( X ) − codim ⁡ ( X ) + 1 reg(X) <= deg(X)-codim(X)+1\operatorname{reg}(X) \leq \operatorname{deg}(X)-\operatorname{codim}(X)+1reg⁡(X)≤deg⁡(X)−codim⁡(X)+1
which is sharp as equality holds for the twisted cubic curve. This is called the Regularity Conjecture. In particular, it yields the following regularity conjecture for prime ideals:
Conjecture 6.4 (Eisenbud-Goto [53], 1984). If L L LLL is a homogeneous prime ideal in S S SSS, and L ( x 1 , , x n ) 2 L ⊂ x 1 , … , x n 2 L sub(x_(1),dots,x_(n))^(2)L \subset\left(x_{1}, \ldots, x_{n}\right)^{2}L⊂(x1,…,xn)2, then
reg ( L ) deg ( L ) reg ⁡ ( L ) ≤ deg ⁡ ( L ) reg(L) <= deg(L)\operatorname{reg}(L) \leq \operatorname{deg}(L)reg⁡(L)≤deg⁡(L)
In particular, L L LLL is generated in degrees deg ( L ) ≤ deg ⁡ ( L ) <= deg(L)\leq \operatorname{deg}(L)≤deg⁡(L).
The condition L ( x 1 , , x n ) 2 L ⊂ x 1 , … , x n 2 L sub(x_(1),dots,x_(n))^(2)L \subset\left(x_{1}, \ldots, x_{n}\right)^{2}L⊂(x1,…,xn)2 is equivalent to requiring that the projective variety V ( L ) V ( L ) V(L)V(L)V(L) is not contained in a hyperplane in P n 1 P n − 1 P^(n-1)\mathbb{P}^{n-1}Pn−1. Prime ideals that satisfy this condition are called nondegenerate.
The Regularity Conjecture is proved for curves by Gruson-Lazarsfeld-Peskine [69], completing fundamental work of Castelnuovo [22]; see also [67]. It is also proved for smooth surfaces by Lazarsfeld [85] and Pinkham [103]. In the smooth case, Kwak [81-83] gives bounds for regularity in dimensions 3 and 4 that are only slightly worse than the optimal ones. The conjecture also holds in the Cohen-Macaulay case by a result of Eisenbud-Goto [53]. Many other special cases and related bounds have been proved as well.
In [92] Jason McCullough and I construct counterexamples to the Regularity Conjecture. We provide a family of prime ideals P r P r P_(r)P_{r}Pr, depending on a parameter r r rrr, whose degree is singly exponential in r r rrr and whose regularity is doubly exponential in r r rrr. Our main theorem is much stronger:
Theorem 6.5 (McCullough-Peeva, [92]). The regularity of nondegenerate homogeneous prime ideals is not bounded by any polynomial function of the degree (multiplicity), i.e., for any polynomial f ( x ) R [ x ] f ( x ) ∈ R [ x ] f(x)inR[x]f(x) \in \mathbb{R}[x]f(x)∈R[x] there exists a nondegenerate homogeneous prime ideal Y Y YYY in a standard graded polynomial ring over C C C\mathbb{C}C such that reg ( Y ) > f ( deg ( Y ) ) reg ⁡ ( Y ) > f ( deg ⁡ ( Y ) ) reg(Y) > f(deg(Y))\operatorname{reg}(Y)>f(\operatorname{deg}(Y))reg⁡(Y)>f(deg⁡(Y)).
For this purpose, we introduce in [92] an approach which, starting from a homogeneous ideal I I III, produces a prime ideal P P PPP whose projective dimension, regularity, degree,
dimension, depth, and codimension are expressed in terms of numerical invariants of I I III. Our approach involves two new concepts:
(1) Rees-like algebras (inspired by an example by Hochster published in [11]) which, unlike the standard Rees algebras, have well-structured defining equations and minimal free resolutions;
(2) A step-by-step homogenization technique which, unlike classical homogenization, preserves graded Betti numbers.
Further research in this direction was carried out by Caviglia-Chardin-McCullough-Peeva-Varbaro in [24]. Our expository paper [93] provides an overview of counterexamples and the techniques used to prove them.
The bound in the Regularity Conjecture is very elegant, so it is reasonable to expect that work will continue on whether it holds when we impose extra conditions on the prime ideal: for example, for smooth varieties or for toric ideals (in the sense of the definition in [101, SECTION 65]).
Instead of trying to repair the Regularity Conjecture by imposing extra conditions, one may wonder:
Question 6.6 (McCullough-Peeva, [93]). What is an optimal function f ( x ) f ( x ) f(x)f(x)f(x) such that reg ( L ) f ( deg ( L ) ) reg ⁡ ( L ) ≤ f ( deg ⁡ ( L ) ) reg(L) <= f(deg(L))\operatorname{reg}(L) \leq f(\operatorname{deg}(L))reg⁡(L)≤f(deg⁡(L)) for any nondegenerate homogeneous prime ideal L L LLL in a standard graded polynomial ring over C C C\mathbb{C}C ?
Since Theorem 5.2 gives a doubly exponential bound on regularity for all homogeneous ideals, and in view of Theorem 6.5, the following question is of interest:
Question 6.7 (McCullough-Peeva, [93]). Does there exist a singly exponential bound for regularity of homogeneous nondegenerate prime ideals in a standard graded polynomial ring over C C C\mathbb{C}C, in terms of the multiplicity alone?
In [8, COMMENTS AFTER THEOREM 3.12] Bayer and Mumford wrote:
"We would conjecture that if a linear bound doesn't hold, at the least a single exponential bound, i.e. reg ( L ) maxdeg ( L ) O ( n ) reg ⁡ ( L ) ≤ maxdeg ⁡ ( L ) O ( n ) reg(L) <= maxdeg(L)^(O(n))\operatorname{reg}(L) \leq \operatorname{maxdeg}(L)^{\mathcal{O}(n)}reg⁡(L)≤maxdeg⁡(L)O(n), ought to hold for any reduced equidimensional ideal. This is an essential ingredient in analyzing the worst-case behavior of all algorithms based on Gröbner bases."
For prime ideals, their conjecture is:
Conjecture 6.8 (Bayer-Mumford, [8, COMMENTS AFTER THEOREM 3.12]). If L L LLL is a homogeneous non-degenerate prime ideal in S = C [ x 1 , , x n ] S = C x 1 , … , x n S=C[x_(1),dots,x_(n)]S=\mathbb{C}\left[x_{1}, \ldots, x_{n}\right]S=C[x1,…,xn], then
reg ( L ) maxdeg ( L ) O ( n ) reg ⁡ ( L ) ≤ maxdeg ⁡ ( L ) O ( n ) reg(L) <= maxdeg(L)^(O(n))\operatorname{reg}(L) \leq \operatorname{maxdeg}(L)^{\mathcal{O}(n)}reg⁡(L)≤maxdeg⁡(L)O(n)
where maxdeg ( L ) maxdeg ⁡ ( L ) maxdeg(L)\operatorname{maxdeg}(L)maxdeg⁡(L) is the maximal degree of an element in a minimal system of homogeneous generators of L L LLL.

7. REGULARITY OF THE RADICAL

Ravi [104] proved that in some cases the regularity of the radical of an ideal is no greater than the regularity of the ideal itself. For a long time, there was a folklore conjecture that this would hold for every homogeneous ideal. However, counterexamples were constructed by Chardin-D'Cruz [40]. They obtained examples where regularity of the radical is nearly the square (or the cube) of that of the ideal.
Theorem 7.1 (Chardin-D'Cruz, [40, EXAMPLE 2.5]). For m 1 m ≥ 1 m >= 1m \geq 1m≥1 and r 3 r ≥ 3 r >= 3r \geq 3r≥3, the ideal
J m , r = ( y m u 2 x m z v , z r + 1 x u r , u r + 1 x v r , y m v r x m 1 z u r 1 v ) J m , r = y m u 2 − x m z v , z r + 1 − x u r , u r + 1 − x v r , y m v r − x m − 1 z u r − 1 v J_(m,r)=(y^(m)u^(2)-x^(m)zv,z^(r+1)-xu^(r),u^(r+1)-xv^(r),y^(m)v^(r)-x^(m-1)zu^(r-1)v)J_{m, r}=\left(y^{m} u^{2}-x^{m} z v, z^{r+1}-x u^{r}, u^{r+1}-x v^{r}, y^{m} v^{r}-x^{m-1} z u^{r-1} v\right)Jm,r=(ymu2−xmzv,zr+1−xur,ur+1−xvr,ymvr−xm−1zur−1v)
in the polynomial ring C [ x , y , z , u , v ] C [ x , y , z , u , v ] C[x,y,z,u,v]\mathbb{C}[x, y, z, u, v]C[x,y,z,u,v] has
reg ( J m , r ) = m + 2 r + 1 reg ( J m , r ) = m ( r 2 2 r 1 ) + 1 reg ⁡ J m , r = m + 2 r + 1 reg ⁡ J m , r = m r 2 − 2 r − 1 + 1 {:[reg(J_(m,r))=m+2r+1],[reg(sqrt(J_(m,r)))=m(r^(2)-2r-1)+1]:}\begin{aligned} \operatorname{reg}\left(J_{m, r}\right) & =m+2 r+1 \\ \operatorname{reg}\left(\sqrt{J_{m, r}}\right) & =m\left(r^{2}-2 r-1\right)+1 \end{aligned}reg⁡(Jm,r)=m+2r+1reg⁡(Jm,r)=m(r2−2r−1)+1
The existence of a polynomial bound is very unclear, so perhaps it is reasonable to focus on the following folklore question which is currently open:
Question 7.2. Is there a singly exponential bound on reg ( I ) ( I ) (sqrtI)(\sqrt{I})(I) in terms of reg(I) (and possibly codim ( I ) ( I ) (I)(I)(I) or n ) n ) n)n)n) for every homogeneous ideal I I III in a standard graded polynomial ring over C C C\mathbb{C}C ?
In order to form reasonable conjectures, it would be very helpful to develop methods for producing interesting examples. In [86, REMARK 1.8.33] Lazarsfeld wrote:
“...the absence of systematic techniques for constructing examples is one of the biggest lacunae in the current state of the theory."

8. SHIFTS

Let T T TTT be a graded finitely generated S S SSS-module. The (upper) shifts are refinements of the numerical invariant regularity. The (upper) shift at step i i iii is
t i ( T ) = max { j b i , j ( T ) 0 } t i ( T ) = max j ∣ b i , j ( T ) ≠ 0 t_(i)(T)=max{j∣b_(i,j)(T)!=0}t_{i}(T)=\max \left\{j \mid b_{i, j}(T) \neq 0\right\}ti(T)=max{j∣bi,j(T)≠0}
and the adjusted shift is
r i ( T ) = max { j b i , i + j ( T ) 0 } r i ( T ) = max j ∣ b i , i + j ( T ) ≠ 0 r_(i)(T)=max{j∣b_(i,i+j)(T)!=0}r_{i}(T)=\max \left\{j \mid b_{i, i+j}(T) \neq 0\right\}ri(T)=max{j∣bi,i+j(T)≠0}
so
r i ( T ) = t i ( T ) i r i ( T ) = t i ( T ) − i r_(i)(T)=t_(i)(T)-ir_{i}(T)=t_{i}(T)-iri(T)=ti(T)−i
Note that r 0 ( T ) r 0 ( T ) r_(0)(T)r_{0}(T)r0(T) is the maximal degree of an element in a minimal system of generators of T T TTT, and
reg ( T ) = max i { r i ( T ) } reg ⁡ ( T ) = max i   r i ( T ) reg(T)=max_(i){r_(i)(T)}\operatorname{reg}(T)=\max _{i}\left\{r_{i}(T)\right\}reg⁡(T)=maxi{ri(T)}
Let L L LLL be a graded ideal in S S SSS. The a , b a , b a,ba, ba,b-subadditivity condition for L L LLL is
(8.1) t a + b ( S / L ) t a ( S / L ) + t b ( S / L ) (8.1) t a + b ( S / L ) ≤ t a ( S / L ) + t b ( S / L ) {:(8.1)t_(a+b)(S//L) <= t_(a)(S//L)+t_(b)(S//L):}\begin{equation*} t_{a+b}(S / L) \leq t_{a}(S / L)+t_{b}(S / L) \tag{8.1} \end{equation*}(8.1)ta+b(S/L)≤ta(S/L)+tb(S/L)
Note that it is equivalent to
r a + b ( S / L ) r a ( S / L ) + r b ( S / L ) r a + b ( S / L ) ≤ r a ( S / L ) + r b ( S / L ) r_(a+b)(S//L) <= r_(a)(S//L)+r_(b)(S//L)r_{a+b}(S / L) \leq r_{a}(S / L)+r_{b}(S / L)ra+b(S/L)≤ra(S/L)+rb(S/L)
We say that L L LLL satisfies the general subadditivity condition if (8.1) holds for every a , b a , b a,ba, ba,b. We say that L L LLL satisfies the initial subadditivity condition if (8.1) holds for b = 1 b = 1 b=1b=1b=1 and every a a aaa. We say that L L LLL satisfies the closing subadditivity condition if (8.1) holds for every a , b a , b a,ba, ba,b with a + b = pd ( L ) a + b = pd ⁡ ( L ) a+b=pd(L)a+b=\operatorname{pd}(L)a+b=pd⁡(L). Gorenstein ideals failing the subadditivity condition were constructed by McCullough-Seceleanu in [95].
Problem 8.1. (1) (McCullough, [91]) It is expected that the general subadditivity condition holds for every monomial ideal L L LLL.
(2) (Avramov-Conca-Iyengar, [7]) It is conjectured that the general subadditivity condition holds if S / L S / L S//LS / LS/L is a Koszul algebra.
(3) (McCullough, [91]) It is expected that the general subadditivity condition holds for every toric ideal L L LLL.
There are supporting results in special cases; the expository paper [91] provides a nice overview of the current state of these problems. For monomial ideals, HerzogSrinivasan [75] proved that the initial subadditivity condition holds.
Another interesting direction of using shifts is:
Problem 8.2. Find good upper bounds on regularity using the shifts in part of the minimal free resolution.
The following result shows how this may work:
Theorem 8.3 (McCullough, [90]). Let L L LLL be a homogeneous ideal in S S SSS. Set c = n 2 c = n 2 c=|~(n)/(2)~|c=\left\lceil\frac{n}{2}\right\rceilc=⌈n2⌉. Then
reg ( S / L ) i = 1 c t i ( S / I ) + i = 1 h t i ( S / I ) ( c 1 ) ! reg ⁡ ( S / L ) ≤ ∑ i = 1 c   t i ( S / I ) + ∏ i = 1 h   t i ( S / I ) ( c − 1 ) ! reg(S//L) <= sum_(i=1)^(c)t_(i)(S//I)+(prod_(i=1)^(h)t_(i)(S//I))/((c-1)!)\operatorname{reg}(S / L) \leq \sum_{i=1}^{c} t_{i}(S / I)+\frac{\prod_{i=1}^{h} t_{i}(S / I)}{(c-1)!}reg⁡(S/L)≤∑i=1cti(S/I)+∏i=1hti(S/I)(c−1)!

9. THE EGH CONJECTURE

We start with a brief introduction to Hilbert functions and lex ideals. If I I III is a homogeneous ideal in S S SSS, then the quotient R := S / I R := S / I R:=S//IR:=S / IR:=S/I inherits the grading by R i = S i / I i R i = S i / I i R_(i)=S_(i)//I_(i)R_{i}=S_{i} / I_{i}Ri=Si/Ii for all i i iii. The size of a homogeneous ideal J J JJJ in R R RRR is measured by its Hilbert function
Hilb R / J ( i ) = dim C ( R i / J i ) for i Z Hilb R / J ⁡ ( i ) = dim C ⁡ R i / J i  for  i ∈ Z Hilb_(R//J)(i)=dim_(C)(R_(i)//J_(i))quad" for "i inZ\operatorname{Hilb}_{R / J}(i)=\operatorname{dim}_{\mathbb{C}}\left(R_{i} / J_{i}\right) \quad \text { for } i \in \mathbf{Z}HilbR/J⁡(i)=dimC⁡(Ri/Ji) for i∈Z
Hilbert's insight was that Hilb R / J Hilb R / J Hilb_(R//J)\operatorname{Hilb}_{R / J}HilbR/J is determined by finitely many of its values. He proved that there exists a polynomial (called the Hilbert polynomial) g ( t ) Q [ t ] g ( t ) ∈ Q [ t ] g(t)inQ[t]g(t) \in \mathbf{Q}[t]g(t)∈Q[t] such that
Hilb R / J ( i ) = g ( i ) for i 0 Hilb R / J ⁡ ( i ) = g ( i )  for  i ≫ 0 Hilb_(R//J)(i)=g(i)quad" for "i≫0\operatorname{Hilb}_{R / J}(i)=g(i) \quad \text { for } i \gg 0HilbR/J⁡(i)=g(i) for i≫0
If S / J S / J S//JS / JS/J (here R = S R = S R=SR=SR=S ) is the coordinate ring of a projective algebraic variety X X XXX, then the degree of the Hilbert polynomial equals the dimension of X X XXX; the leading coefficient of the Hilbert polynomial determines another important invariant - the degree (multiplicity) of X X XXX. Hilbert functions for monomial ideals in the ring C [ x 1 , , x n ] / ( x 1 2 , , x n 2 ) C x 1 , … , x n / x 1 2 , … , x n 2 C[x_(1),dots,x_(n)]//(x_(1)^(2),dots,x_(n)^(2))\mathbb{C}\left[x_{1}, \ldots, x_{n}\right] /\left(x_{1}^{2}, \ldots, x_{n}^{2}\right)C[x1,…,xn]/(x12,…,xn2) have been extensively studied in Combinatorics since each such Hilbert function counts the number of faces in a simplicial complex.
Lex ideals are fruitful tools in the study of Hilbert functions. They are monomial ideals defined in a simple way: Denote by > lex > lex  > _("lex ")>_{\text {lex }}>lex  the lexicographic order on the monomials in S S SSS extending x 1 > > x n x 1 > ⋯ > x n x_(1) > cdots > x_(n)x_{1}>\cdots>x_{n}x1>⋯>xn. A monomial ideal L L LLL in S S SSS is lex if the following property holds: if m L m ∈ L m in Lm \in Lm∈L is a monomial and q > lex m q > lex  m q > _("lex ")mq>_{\text {lex }} mq>lex m is a monomial of the same degree, then q L q ∈ L q in Lq \in Lq∈L (that is, for each i 0 i ≥ 0 i >= 0i \geq 0i≥0 the vector space L i L i L_(i)L_{i}Li is either zero or is spanned by lex-consecutive monomials of degree i i iii starting with x 1 i x 1 i x_(1)^(i)x_{1}^{i}x1i ).
A core result in Commutative Algebra is Macaulay's Theorem 9.1, which characterizes the Hilbert functions of homogeneous ideals in the polynomial ring S S SSS :
Theorem 9.1 (Macaulay, [87]). For every homogeneous ideal in S S SSS there exists a lex ideal with the same Hilbert function.
The Hilbert function of a lex ideal is easy to count. This leads to an equivalent formulation of Macaulay's Theorem 9.1 which characterizes numerically (by certain inequalities) the Hilbert functions of homogeneous ideals; see [101, SECTION 49].
Lex ideals also play an important role in the study of Hilbert schemes. Grothendieck introduced the Hilbert scheme H r , g H r , g H_(r,g)\mathscr{H}_{r, g}Hr,g that parametrizes subschemes of P r P r P^(r)\mathbf{P}^{r}Pr with a fixed Hilbert polynomial g g ggg. The structure of the Hilbert scheme is known to be very complicated. In [71] Harris and Morrison state Murphy's Law for Hilbert schemes:
"There is no geometric possibility so horrible that it cannot be found generically on some component of some Hilbert scheme."
The main structural result on H r , g H r , g H_(r,g)\mathscr{H}_{r, g}Hr,g is Hartshorne's Theorem:
Theorem 9.2 (Hartshorne, [72]). The Hilbert scheme H r , g H r , g H_(r,g)\mathscr{H}_{r, g}Hr,g is connected.
The situation is that every homogeneous ideal with a fixed Hilbert function h h hhh is connected by a sequence of deformations to the lex ideal with Hilbert function h h hhh. A deformation connects two ideals J t = 0 J t = 0 J_(t=0)J_{t=0}Jt=0 and J t = 1 J t = 1 J_(t=1)J_{t=1}Jt=1 in the sense that we have a family of homogeneous ideals J t J t J_(t)J_{t}Jt varying with the parameter t [ 0 , 1 ] t ∈ [ 0 , 1 ] t in[0,1]t \in[0,1]t∈[0,1] so that the Hilbert function is preserved; in this case, the ideals J t J t J_(t)J_{t}Jt form a path on the Hilbert scheme. Hartshorne's proof [72] relies on deformations called "distractions" which use generic change of coordinates and polarization. Analyzing the paths on a Hilbert scheme may shed light on whether there exists an object with maximal Betti numbers.
Theorem 9.3 (Bigatti-Hulett-Pardue, see [100]). A lex ideal in S S SSS has the greatest Betti numbers among all homogeneous ideals in S S SSS with the same Hilbert function.
This result was quite surprising when it was discovered since counterexamples were known in which no ideal with a fixed Hilbert function attains minimal Betti numbers. It yields numerical upper bounds on Betti numbers as follows: the minimal free resolution of a lex ideal is the Eliahou-Kervaire resolution [57] (see [101, SEcTION 28]), and it provides numerical formulas for the Betti numbers of a lex ideal.
It is natural to ask if similar results hold over other rings. For starters, we need rings over which Theorem 9.1 holds. It actually fails over most graded quotient rings of S S SSS. For example, there is no lex ideal in the ring C [ x , y ] / ( x 2 y , x y 2 ) C [ x , y ] / x 2 y , x y 2 C[x,y]//(x^(2)y,xy^(2))\mathbb{C}[x, y] /\left(x^{2} y, x y^{2}\right)C[x,y]/(x2y,xy2) with the same Hilbert function as the ideal ( x y ) ( x y ) (xy)(x y)(xy).
Theorem 9.4. Macaulay's Theorem 9.1 holds over the following rings:
(1) (Kruskal, Katona, [77,80]) an exterior algebra E over C C C\mathbb{C}C.
(2) (Clements-Lindström, [45]) a Clements-Lindström ring C := C [ x 1 , , x n ] / P C := C x 1 , … , x n / P C:=C[x_(1),dots,x_(n)]//PC:=\mathbb{C}\left[x_{1}, \ldots, x_{n}\right] / PC:=C[x1,…,xn]/P, where P P PPP is an ideal generated by powers of the variables.
(3) (Gasharov-Murai-Peeva, [66]) a Veronese ring V := S / J V := S / J V:=S//JV:=S / JV:=S/J, where J J JJJ is the defining ideal of a Veronese toric variety.
Proving analogues of Theorem 9.3 for the above rings is difficult since minimal resolutions over exterior algebras, Clements-Lindström rings, or Veronese rings are infinite (in contrast, Theorem 9.3 is about finite resolutions) and so they are considerably more intricate. It was proved that every lex ideal has the greatest Betti numbers among all homogeneous ideals with the same Hilbert function over the following rings: over E E EEE by AramovaHerzog-Hibi [6], over C C CCC by Murai-Peeva [99], and over V V VVV by Gasharov-Murai-Peeva [66].
Hilbert functions of ideals containing ( x 1 2 , , x n 2 ) x 1 2 , … , x n 2 (x_(1)^(2),dots,x_(n)^(2))\left(x_{1}^{2}, \ldots, x_{n}^{2}\right)(x12,…,xn2) are characterized numerically (by certain inequalities) by Kruskal-Katona's Theorem [77,80], which is a natural analogue of Macaulay's Theorem; see Theorem 9.4(1,2). Eisenbud-Green-Harris conjectured that the same numerical inequalities for the Hilbert function hold for all ideals in S S SSS containing a quadratic regular sequence:
Conjecture 9.5 (Eisenbud-Green-Harris, [54]). Let L S L ⊂ S L sub SL \subset SL⊂S be a homogeneous ideal containing a regular sequence of n n nnn quadratic forms. There exists an ideal N N NNN containing x 1 2 , , x n 2 x 1 2 , … , x n 2 x_(1)^(2),dots,x_(n)^(2)x_{1}^{2}, \ldots, x_{n}^{2}x12,…,xn2 with the same Hilbert function as L L LLL.
Kruskal-Katona's Theorem was generalized by Clements-Lindström [45] to a characterization of the Hilbert functions of ideals containing powers of the variables; see Theorem 9.4(2). In view of this, Eisenbud-Green-Harris noted in [54] that Conjecture 9.5 can be extended to cover all complete intersections as follows:
Conjecture 9.6 (Eisenbud-Green-Harris, [54]). Let L S L ⊂ S L sub SL \subset SL⊂S be a homogeneous ideal containing a regular sequence of forms of degrees a 1 a n a 1 ≤ ⋯ ≤ a n a_(1) <= cdots <= a_(n)a_{1} \leq \cdots \leq a_{n}a1≤⋯≤an. There exists an ideal N N NNN containing x 1 a 1 , , x n a n x 1 a 1 , … , x n a n x_(1)^(a_(1)),dots,x_(n)^(a_(n))x_{1}^{a_{1}}, \ldots, x_{n}^{a_{n}}x1a1,…,xnan with the same Hilbert function as L L LLL.
Conjecture 9.5 is considered to be the main case of the Eisenbud-Green-Harris Conjectures, called the EGH Conjectures.
In their original form in [54], the EGH Conjectures are stated in terms of numerical inequalities for the Hilbert function. We give an equivalent form, which follows immediately from the Clements-Lindström Theorem 9.4(2).
Eisenbud, Green, and Harris were led to the EGH Conjectures by extending a series of results and conjectures in Castelnuovo Theory in [54]. After that, they made the connection to the Cayley-Bacharach Theory in [55]. They provide in [55] a nice survey of the long history of Cayley-Bacharach theory in Algebraic Geometry.
The EGH Conjectures turned out to be very challenging. Some special cases, applications, and related results are proved in [ 2 , 3 , 25 , 27 , 30 , 33 , 34 , 43 , 44 , 46 , 47 , 54 , 55 , 59 , 70 , 96 , 105 , 106 ] [ 2 , 3 , 25 , 27 , 30 , 33 , 34 , 43 , 44 , 46 , 47 , 54 , 55 , 59 , 70 , 96 , 105 , 106 ] [2,3,25,27,30,33,34,43,44,46,47,54,55,59,70,96,105,106][2,3,25,27,30,33,34,43,44,46,47,54,55,59,70,96,105,106][2,3,25,27,30,33,34,43,44,46,47,54,55,59,70,96,105,106]. One of the strongest results is the recent paper [26] by Caviglia-DeStefani.
We now focus on Betti numbers related to the EGH Conjectures. Let L S L ⊂ S L sub SL \subset SL⊂S be a homogeneous ideal containing a regular sequence of forms of degrees a 1 a n a 1 ≤ ⋯ ≤ a n a_(1) <= cdots <= a_(n)a_{1} \leq \cdots \leq a_{n}a1≤⋯≤an. The concept of a lex ideal can be generalized to the concept of a lex-plus-powers ideal which is a monomial ideal containing x 1 a 1 , , x n a n x 1 a 1 , … , x n a n x_(1)^(a_(1)),dots,x_(n)^(a_(n))x_{1}^{a_{1}}, \ldots, x_{n}^{a_{n}}x1a1,…,xnan and otherwise is like a lex ideal. G. Evans conjectured the more general Lex-Plus-Powers Conjecture that, among all graded ideals with a fixed Hilbert function and containing a homogeneous regular sequence of degrees a 1 a 1 ≤ a_(1) <=a_{1} \leqa1≤ a n ⋯ ≤ a n cdots <= a_(n)\cdots \leq a_{n}⋯≤an, the lex-plus-powers ideal (which exists according to the EGH Conjectures) has the greatest Betti numbers. This conjecture was inspired by Theorem 9.3.
Theorem 9.7 (Mermin-Murai, [97]). The Lex-Plus-Powers Conjecture holds for ideals containing pure powers.
The general Lex-Plus-Powers Conjecture (for ideals containing a homogeneous regular sequence) is very difficult. Some special cases are proved in [ 31 , 59 , 63 , 64 , 105 , 106 ] [ 31 , 59 , 63 , 64 , 105 , 106 ] [31,59,63,64,105,106][31,59,63,64,105,106][31,59,63,64,105,106]. The expository papers by Caviglia-DeStefani-Sbarra [28] and by Francisco-Richert [65] provide nice overviews of this challenging topic.

FUNDING

This work was partially supported by NSF grant DMS-2001064.

REFERENCES

[1] J. Abbott, A. Bigatti, and L. Robbiano, CoCoA: a system for doing computations in Commutative Algebra, available at http://cocoa.dima.unige.it.
[2] A. Abedelfatah, On the Eisenbud-Green-Harris conjecture. Proc. Amer. Math. Soc. 143 (2015), 105-115.
[3] A. Abedelfatah, Hilbert functions of monomial ideals containing a regular sequence. Israel J. Math. 214 (2016), 857-865.
[4] T. Ananyan and M. Hochster, Small subalgebras of polynomial rings and Stillman's Conjecture. J. Amer. Math. Soc. 33 (2019), 291-309.
[5] M. Aprodu, G. Farkas, S. Papadima, C. Raicu, and J. Weyman, Koszul modules and Green's conjecture. Invent. Math. 218 (2019), 657-720.
[6] A. Aramova, J. Herzog, and T. Hibi, Gotzmann theorems for exterior algebras and combinatorics. J. Algebra 191 (1997), 174-211.
[7] L. Avramov, A. Conca, and S. Iyengar, Subadditivity of syzygies of Koszul algebras. Math. Ann. 361 (2015), 511-534.
[8] D. Bayer and D. Mumford, What can be computed in algebraic geometry? In Computational algebraic geometry and commutative algebra. Symposia Mathematica XXXIV, pp. 1-48, Cambridge University Press, Cambridge, 1993.
[9] D. Bayer and M. Stillman, A criterion for detecting m m mmm-regularity. Invent. Math. 87 (1987), 1-11.
[10] D. Bayer and M. Stillman, On the complexity of computing syzygies. Computational aspects of commutative algebra. J. Symbolic Comput. 6 (1988), 135-147.
[11] J. Becker, On the boundedness and the unboundedness of the number of generators of ideals and multiplicity. J. Algebra 48 (1977), 447-453.
[12] J. Beder, J. McCullough, L. Núnez-Betancourt. A. Seceleanu, B. Snapp, and B. Stone, Ideals with larger projective dimension and regularity. J. Symbolic Comput. 46 (2011), 1105-1113.
[13] C. Berkesch, D. Erman, and G. Smith, Virtual resolutions for a product of projective spaces. Algebr. Geom. 7 (2020), 460-481.
[14] C. Berkesch, M. Kummini, D. Erman, and S. Sam, Poset structures in BoijSöderberg theory. J. Eur. Math. Soc. 15 (2013), 2257-2295.
[15] A. Bertram, L. Ein, and R. Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties. J. Amer. Math. Soc. 4 (1991), 587-602.
[16] M. Boij and J. Söderberg, Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture. J. Lond. Math. Soc. 78 (2008), 85-106.
[17] A. Boocher and E. Grifo, Lower bounds on Betti numbers. In Commutative algebra: Expository papers in honor of David Eisenbud's 75'th birthday, edited by I. Peeva, Springer, 2022
[18] W. Bruns, "Jede" endliche freie Auflösung ist freie Auflösung eines von drei Elementen erzeugten Ideals. J. Algebra 39 (1976), 429-439.
[19] W. Bruns, A. Conca, and M. Varbaro, Castelnuovo-Mumford regularity and powers. In Commutative algebra: Expository papers in honor of David Eisenbud's 75 'th birthday, edited by I. Peeva, Springer, 2022.
[20] D. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. Amer. J. Math. 99 (1977), 447-485.
[21] L. Burch, A note on the homology of ideals generated by three elements in local rings. Proc. Camb. Philos. Soc. 64 (1968), 949-952.
[22] G. Castelnuovo, Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica. Rend. Circ. Mat. Palermo 7 (1893), 89-110.
[23] G. Caviglia, Koszul algebras, Castelnuovo-Mumford regularity and generic initial ideals. PhD Thesis, University of Kansas, 2004.
[24] G. Caviglia, M. Chardin, J. McCullough, I. Peeva, and M. Varbaro, Regularity of prime ideals. Math. Z. 291 (2019), 421-435.
[25] G. Caviglia, A. Constantinescu, and M. Varbaro, On a conjecture by Kalai. Israel J. Math. 204 (2014), 469-475.
[26] G. Caviglia and A. De Stefani, A Cayley-Bacharach theorem for points in P n P n P^(n)\mathbf{P}^{n}Pn. 2020, arXiv:2006.14717.
[27] G. Caviglia and A. De Stefani, The Eisenbud-Green-Harris conjecture for fastgrowing degree sequences. 2020, arXiv:2007.15467.
[28] G. Caviglia, A. De Stefani, and E. Sbarra, The Eisenbud-Green-Harris conjecture. In Commutative algebra: Expository papers in honor of David Eisenbud's 75 'th birthday, edited by I. Peeva, Springer, 2022.
[29] G. Caviglia and Y. Liang, Explicit Stillman bounds for all degrees. 2020, arXiv:2009.02826.
[30] G. Caviglia and D. Maclagan, Some cases of the Eisenbud-Green-Harris conjecture. Math. Res. Lett. 15 (2008), 427-433.
[31] G. Caviglia and A. Sammartano, On the lex-plus-powers conjecture. Adv. Math. 340 (2018), 284-299.
[32] G. Caviglia and E. Sbarra, Characteristic-free bounds for the CastelnuovoMumford regularity. Compos. Math. 141 (2005), 1365-1373.
[33] G. Caviglia and E. Sbarra, Distractions of Shakin rings. J. Algebra 419 (2014), 318-331.
[34] G. Caviglia and E. Sbarra, The lex-plus-powers inequality for local cohomology modules. Math. Ann. 364 (2016), 225-241.
[35] A. Cayley, On the theory of elimination. Cambridge and Dublin Math. J. 3 (1848), 116-120; Collected papers Vol. I, pp. 116-120, Cambridge University Press, 1889 .
[36] H. Charalambous and G. Evans, Problems on Betti numbers of finite length modules. In Free resolutions in commutative algebra and algebraic geometry (Sundance, UT, 1990), pp. 25-33, Res. Notes Math. 2, Jones and Bartlett, Boston, 1992.
[37] H. Charalambous, G. Evans, and M. Miller, Betti numbers for modules of finite length. Proc. Amer. Math. Soc. 109 (1990), 63-70.
[38] M. Chardin, Bounds for Castelnuovo-Mumford regularity in terms of degrees of defining equations. In Commutative algebra, singularities and computer algebra (Sinaia, 2002), pp. 67-73, NATO Sci. Ser. II Math. Phys. Chem. 115, Kluwer Acad. Publ., Dordrecht, 2003.
[39] M. Chardin, Some results and questions on Castelnuovo-Mumford regularity. In Syzygies and Hilbert functions. Lect. Notes Pure Appl. Math. 254, pp. 1-40, Chapman & Hall/CRC, Boca Raton, FL, 2007.
[40] M. Chardin and C. D'Cruz, Castelnuovo-Mumford regularity: Examples of curves and surfaces. J. Algebra 270 (2003), 347-360.
[41] M. Chardin and A. Fall, Sur la régularité de Castelnuovo-Mumford des idéaux, en dimension 2. C. R. Math. Acad. Sci. Paris 341 (2005), 233-238.
[42] M. Chardin and B. Ulrich, Liaison and Castelnuovo-Mumford regularity. Amer. J. Math. 124 (2002), 1103-1124.
[43] R.-X. Chen, Some special cases of the Eisenbud-Green-Harris conjecture. Illinois J. Math. 56 (2012), 661-675.
[44] K. Chong, An application of liaison theory to the Eisenbud-Green-Harris conjecture. J. Algebra 445 (2016), 221-231.
[45] G. Clements and B. Lindström, A generalization of a combinatorial theorem of Macaulay. J. Combin. Theory 7 (1969), 230-238.
[46] S. Cooper, Growth conditions for a family of ideals containing regular sequences. J. Pure Appl. Algebra 212 (2008), 122-131.
[47] S. Cooper, Subsets of complete intersections and the EGH conjecture. In Progress in commutative algebra 1, pp. 167-198, de Gruyter, Berlin, 2012.
[48] T. de Fernex and L. Ein, A vanishing theorem for log canonical pairs. Amer. J. Math. 132 (2010), 1205-1221.
[49] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 4-1-2 - A computer algebra system for polynomial computations, available at http://www. singular.uni-kl.de, 2019.
[50] J. Draisma, M. Lason, and A. Leykin, Stillman's conjecture via generic initial ideals. Comm. Algebra 47 (2019), 2384-2395.
[51] D. Eisenbud, The geometry of syzygies. A second course in commutative algebra and algebraic geometry. Grad. Texts in Math. 229, Springer, 2005.
[52] D. Eisenbud, G. Fløystad, and J. Weyman, The existence of equivariant pure free resolutions. Ann. Inst. Fourier 61 (2011), 905-926.
[53] D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity. J. Algebra 88 (1984), 89-133.
[54] D. Eisenbud, M. Green, and J. Harris, Higher Castelnuovo theory. In Journées de Géométrie Algébrique d'Orsay (Orsay, 1992), Astérisque 218 (1993), 187-202.
[55] D. Eisenbud, M. Green, and J. Harris, Cayley-Bacharach theorems and conjectures. Bull. Amer. Math. Soc. (N.S.) 33 (1996), 295-324.
[56] D. Eisenbud and F.-O. Schreyer, Betti numbers of graded modules and cohomology of vector bundles. J. Amer. Math. Soc. 22 (2009), 859-888.
[57] S. Eliahou and M. Kervaire, Minimal resolutions of some monomial ideals. J. Algebra 129 (1990), 1-25.
[58] D. Erman, S. Sam, and A. Snowden, Big polynomial rings and Stillman's conjecture. Invent. Math. 218 (2019), 413-439.
[59] G. Evans and E. Riehl, On the intersections of polynomials and the CayleyBacharach theorem. J. Pure Appl. Algebra 183 (2003), 293-298.
[60] G. Fløystad, Boij-Söderberg theory: introduction and survey. In Progress in commutative algebra 1, pp. 1-54, de Gruyter, 2012.
[61] G. Fløystad, Zipping Tate resolutions and exterior coalgebras. J. Algebra 437 (2015), 249-307.
[62] G. Fløystad, J. McCullough, and I. Peeva, Three themes of syzygies. Bull. Amer. Math. Soc. (N.S.) 53 (2016), 415-435.
[63] C. Francisco, Almost complete intersections and the lex-plus-powers conjecture. J. Algebra 276 (2004), 737-760.
[64] C. Francisco, Hilbert functions and graded free resolutions. PhD Thesis, Cornell University, 2004.
[65] C. Francisco and B. Richert, Lex-plus-powers ideals. In Syzygies and Hilbert functions, edited by I. Peeva, Lect. Notes Pure Appl. Math. 254, pp. 145-178, CRC Press, 2007.
[66] V. Gasharov, S. Murai, and I. Peeva, Hilbert schemes and maximal Betti numbers over Veronese rings. Math. Z. 267 (2011), 155-172.
[67] D. Giaimo, On the Castelnuovo-Mumford regularity of connected curves. Trans. Amer. Math. Soc. 358 (2006), 267-284.
[68] D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.
[69] L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo and the equations defining projective varieties. Invent. Math. 72 (1983), 491-506.
[70] T. Harima, A. Wachi, and J. Watanabe, The EGH conjecture and the Sperner property of complete intersections. Proc. Amer. Math. Soc. 145 (2017), 1497-1503.
[71] J. Harris and I. Morrison, Moduli of curves. Grad. Texts in Math. 187, Springer, New York, 1998 .
[72] R. Hartshorne, Connectedness of the Hilbert scheme. Publ. Math. Inst. Hautes Études Sci. 29 (1966), 5-48.
[73] R. Hartshorne, Algebraic vector bundles on projective spaces: a problem list. Topology 18 (1979), 117-128.
[74] J. Herzog and M. Kühl, On the Betti numbers of finite pure and linear resolutions. Comm. Algebra 12 (1984), 1627-1646.
[75] J. Herzog and H. Srinivasan, On the subadditivity problem for maximal shifts in free resolutions. In Commutative algebra and noncommutative algebraic geometry, Vol. II, pp. 245-249, Math. Sci. Res. Inst. Publ. 68, Cambridge Univ. Press, New York, 2015.
[76] D. Hilbert, Ueber die Theorie der algebraischen Formen. Math. Ann. 36 (1890), 473-534.
[77] G. Katona, A theorem for finite sets. In Theory of graphs, pp. 187-207, Academic Press, New York, 1968.
[78] J. Koh, Ideals generated by quadrics exhibiting double exponential degrees.
J. Algebra 200 (1998), 225-245.
[79] P. Kohn, Ideals generated by three elements. Proc. Amer. Math. Soc. 35 (1972), 55 58 55 − 58 55-5855-5855−58.
[80] J. Kruskal, The number of simplices in a complex. In Mathematical optimization techniques, edited by R. Bellman, pp. 251-278, University of California Press, Berkeley/Los Angeles, 1963.
[81] S. Kwak, Castelnuovo regularity for smooth subvarieties of dimensions 3 and 4. J. Algebraic Geom. 7 (1998), 195-206.
[82] S. Kwak, Castelnuovo-Mumford regularity bound for smooth threefolds in P 5 P 5 P^(5)\mathbf{P}^{5}P5 and extremal examples. J. Reine Angew. Math. 509 (1999), 21-34.
[83] S. Kwak, Generic projections, the equations defining projective varieties and Castelnuovo regularity. Math. Z. 234 (2000), 413-434.
[84] S. Kwak and J. Park, A bound for Castelnuovo-Mumford regularity by double point divisors. Adv. Math. 364 (2020).
[85] R. Lazarsfeld, A sharp Castelnuovo bound for smooth surfaces. Duke Math. J. 55 (1987), 423-438.
[86] R. Lazarsfeld, Positivity in algebraic geometry. I. II. Ergeb. Math. Grenzgeb. (3) 48, 49, Springer, Berlin, 2004.
[87] F. Macaulay, Some properties of enumeration in the theory of modular systems. Proc. Lond. Math. Soc. 26 (1927), 531-555.
[88] E. Mayr and A. Meyer, The complexity of the word problem for commutative semigroups and polynomial ideals. Adv. Math. 46 (1982), 305-329.
[89] J. McCullough, A family of ideals with few generators in low degree and large projective dimension. Proc. Amer. Math. Soc. 139 (2011), 2017-2023.
[90] J. McCullough, A polynomial bound on the regularity of an ideal in terms of half of the syzygies. Math. Res. Lett. 19 (2012), 555-565.
[91] J. McCullough, Subadditivity of syzygies of ideals and related problems. In Commutative algebra: Expository papers in honor of David Eisenbud's 75 'th birthday, edited by I. Peeva, Springer, 2022.
[92] J. McCullough and I. Peeva, Counterexamples to the Eisenbud-Goto regularity conjecture. J. Amer. Math. Soc. 31 (2018), 473-496.
[93] J. McCullough and I. Peeva, The Regularity Conjecture for prime ideals in polynomial rings. EMS Surv. Math. Sci. 7 (2020), 173-206.
[94] J. McCullough and A. Seceleanu, Bounding projective dimension. In Commutative algebra. Expository papers dedicated to David Eisenbud on the occasion of his 65th birthday, edited by I. Peeva, pp. 551-576, Springer, London, 2013.
[95] J. McCullough and A. Seceleanu, Quadratic Gorenstein algebras with many surprising properties. 2020, arXiv:2004.10237.
[96] J. Mermin, Monomial regular sequences. Proc. Amer. Math. Soc. 138 (2010), 1983 1988 1983 − 1988 1983-19881983-19881983−1988.
[97] J. Mermin and S. Murai, The lex-plus-powers conjecture holds for pure powers. Adv. Math. 226 (2011), 3511-3539.
[98] D. Mumford, Lectures on curves on an algebraic surface, with a section by G. M. Bergman. Ann. of Math. Stud. 59, Princeton University Press, Princeton, N J , 1966 N J , 1966 NJ,1966\mathrm{NJ}, 1966NJ,1966.
[99] S. Murai and I. Peeva, Hilbert schemes and Betti numbers over a ClementsLindström rings. Compos. Math. 148 (2012), 1337-1364.
[100] K. Pardue, Deformation classes of graded modules and maximal Betti numbers. Illinois J. Math. 40 (1996), 564-585.
[101] I. Peeva, Graded syzygies. Springer, London, 2011.
[102] I. Peeva and M. Stillman, Open problems on syzygies and Hilbert functions. J. Commut. Algebra 1 (2009), 159-195.
[103] H. Pinkham, A Castelnuovo bound for smooth surfaces. Invent. Math. 83 (1986), 321-332.
[104] M. Ravi, Regularity of ideals and their radicals. Manuscripta Math. 68 (1990), 77-87.
[105] B. Richert, A proof of Evans' convexity conjecture. Comm. Algebra 43 (2015), 3275-3281.
[106] B. Richert and S. Sabourin, The residuals of lex plus powers ideals and the Eisenbud-Green-Harris conjecture. Illinois J. Math. 52 (2008), 1355-1384.
[107] B. Ullery, Designer ideals with high Castelnuovo-Mumford regularity. Math. Res. Lett. 21 (2014), 1215-1225.
[108] M. Walker, Total Betti numbers of modules of finite projective dimension. Ann. of Math. (2) 186 (2017), 641-646.

IRENA PEEVA

Department of Mathematics, Cornell University, Ithaca, NY 14853, USA,
  1. NUMBER THEORY
SPECIAL LECTURE

SURVEY LECTURE ON ARITHMETIC DYNAMICS

JOSEPH H. SILVERMAN

Abstract

Arithmetic dynamics is a relatively new field in which classical problems from number theory and algebraic geometry are reformulated in the setting of dynamical systems. Thus, for example, rational points on algebraic varieties become rational points in orbits, and torsion points on abelian varieties become points having finite orbits. Moduli problems also appear, where, for example, the complex multiplication points in the moduli space of abelian varieties correspond to the postcritically finite points in the moduli space of rational maps. In this article we give a survey of some of the major problems motivating the field of arithmetic dynamics, and some of the progress that has been made during the past 20 years.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 37P15; Secondary 37P05, 37P30, 37P35, 37P45

KEYWORDS

Dynamical uniform boundedness, dynamical unlikely intersection, arboreal representation, dynamical moduli space, dynamical and arithmetic degree

1. INTRODUCTION

This article is a survey of the comparatively new field of Arithmetic Dynamics, a field where arithmetic and dynamics join forces. 1 1 ^(1){ }^{1}1 But the word "arithmetic" in "arithmetic dynamics" is itself short for "arithmetic geometry," a field where the venerable subjects of number theory and algebraic geometry meet. Thus arithmetic dynamics is a melting pot filled with ingredients from three classical areas of mathematics.
In this article we will discuss arithmetic dynamics over global fields, which for the sake of exposition we will generally take to be number fields, i.e., finite extensions of Q Q Q\mathbb{Q}Q. Our primary focus will be dynamical analogues and generalizations of famous theorems and conjectures in arithmetic geometry, centered around the following five major topics that have helped drive the development of arithmetic dynamics over the past few decades:
  • Topic #1: Dynamical Uniform Boundedness
  • Topic #2: Dynamical Moduli Spaces
  • Topic #3: Dynamical Unlikely Intersections
  • Topic #4: Dynatomic and Arboreal Representations
  • Topic #5: Dynamical and Arithmetic Complexity
Remark 1.1. Of course, our chosen five topics do not fully cover the varied problems that fall under the rubric of arithmetic dynamics over global fields. And there are also highly active areas of arithmetic dynamics in which people study dynamical systems defined over non-archimedean fields such as Q p Q p Q_(p)\mathbb{Q}_{p}Qp and C p C p C_(p)\mathbb{C}_{p}Cp and over finite fields F q F q F_(q)\mathbb{F}_{q}Fq. We refer the interested reader to the survey article [10] for a more extensive discussion. As Jung might have said: "The meeting of two mathematical fields is like the contact of two chemical substances: if there is any reaction, both are transformed."

2. DEFINITIONS AND TERMINOLOGY

An abstract dynamical system is simply an object X X XXX and an endomorphism (selfmap ) 2 ) 2 )^(2))^{2})2
f : X X f : X → X f:X rarr Xf: X \rightarrow Xf:X→X
The iterates of f f fff are denoted by
f n = f f f n copies of f f n = f ∘ f ∘ ⋯ ∘ f ⏟ n  copies of  f f^(n)=ubrace(f@f@cdots@fubrace)_(n" copies of "f)f^{n}=\underbrace{f \circ f \circ \cdots \circ f}_{n \text { copies of } f}fn=f∘f∘⋯∘f⏟n copies of f
and the (forward) f f fff-orbit of an element x X x ∈ X x in Xx \in Xx∈X is its image for the iterates of f , 3 f , 3 f,3f, 3f,3
O f ( x ) = { f n ( x ) : n 0 } O f ( x ) = f n ( x ) : n ≥ 0 O_(f)(x)={f^(n)(x):n >= 0}\mathcal{O}_{f}(x)=\left\{f^{n}(x): n \geq 0\right\}Of(x)={fn(x):n≥0}
We say that x X x ∈ X x in Xx \in Xx∈X is f f fff-periodic if
f n ( x ) = x for some n 1 f n ( x ) = x  for some  n ≥ 1 f^(n)(x)=x quad" for some "n >= 1f^{n}(x)=x \quad \text { for some } n \geq 1fn(x)=x for some n≥1
in which case the smallest such n n nnn is the (exact) period of x x xxx. A point x X x ∈ X x in Xx \in Xx∈X is f f fff-preperiodic if its f f fff-orbit O f ( x ) O f ( x ) O_(f)(x)\mathcal{O}_{f}(x)Of(x) is finite, or equivalently, if f m ( x ) f m ( x ) f^(m)(x)f^{m}(x)fm(x) is periodic for some m 0 m ≥ 0 m >= 0m \geq 0m≥0.
Two dynamical systems f 1 , f 2 : X X f 1 , f 2 : X → X f_(1),f_(2):X rarr Xf_{1}, f_{2}: X \rightarrow Xf1,f2:X→X are isomorphic if there is an automor phism φ Aut ( X ) phism ⁡ φ ∈ Aut ⁡ ( X ) phism varphi in Aut(X)\operatorname{phism} \varphi \in \operatorname{Aut}(X)phism⁡φ∈Aut⁡(X) such that
(2.1) f 2 = f 1 φ = φ 1 f 1 φ (2.1) f 2 = f 1 φ = φ − 1 ∘ f 1 ∘ φ {:(2.1)f_(2)=f_(1)^(varphi)=varphi^(-1)@f_(1)@varphi:}\begin{equation*} f_{2}=f_{1}^{\varphi}=\varphi^{-1} \circ f_{1} \circ \varphi \tag{2.1} \end{equation*}(2.1)f2=f1φ=φ−1∘f1∘φ
Note that (2.1) is a good notion of isomorphism for dynamics, since it respects iteration,
( f φ ) n = ( φ 1 f φ ) n = φ 1 f n φ = ( f n ) φ f φ n = φ − 1 ∘ f ∘ φ n = φ − 1 ∘ f n ∘ φ = f n φ (f^(varphi))^(n)=(varphi^(-1)@f@varphi)^(n)=varphi^(-1)@f^(n)@varphi=(f^(n))^(varphi)\left(f^{\varphi}\right)^{n}=\left(\varphi^{-1} \circ f \circ \varphi\right)^{n}=\varphi^{-1} \circ f^{n} \circ \varphi=\left(f^{n}\right)^{\varphi}(fφ)n=(φ−1∘f∘φ)n=φ−1∘fn∘φ=(fn)φ
In particular, orbits and (pre)periodic points of the isomorphic dynamical systems f f fff and f φ f φ f^(varphi)f^{\varphi}fφ are more-or-less identical, since
O f φ ( x ) = φ 1 ( O f ( φ ( x ) ) ) , Per ( f φ ) = φ 1 ( Per ( f ) ) , PrePer ( f φ ) = φ 1 ( PrePer ( f ) ) O f φ ( x ) = φ − 1 O f ( φ ( x ) ) , Per ⁡ f φ = φ − 1 ( Per ⁡ ( f ) ) , PrePer ⁡ f φ = φ − 1 ( PrePer ⁡ ( f ) ) O_(f^(varphi))(x)=varphi^(-1)(O_(f)(varphi(x))),quad Per(f^(varphi))=varphi^(-1)(Per(f)),quad PrePer(f^(varphi))=varphi^(-1)(PrePer(f))\mathcal{O}_{f^{\varphi}}(x)=\varphi^{-1}\left(\mathcal{O}_{f}(\varphi(x))\right), \quad \operatorname{Per}\left(f^{\varphi}\right)=\varphi^{-1}(\operatorname{Per}(f)), \quad \operatorname{PrePer}\left(f^{\varphi}\right)=\varphi^{-1}(\operatorname{PrePer}(f))Ofφ(x)=φ−1(Of(φ(x))),Per⁡(fφ)=φ−1(Per⁡(f)),PrePer⁡(fφ)=φ−1(PrePer⁡(f)).
We conclude this section with a brief discussion of endomorphisms f : P 1 P 1 f : P 1 → P 1 f:P^(1)rarrP^(1)f: \mathbb{P}^{1} \rightarrow \mathbb{P}^{1}f:P1→P1 i.e., rational functions of one variable. For P P 1 P ∈ P 1 P inP^(1)P \in \mathbb{P}^{1}P∈P1, we choose a local parameter z P z P z_(P)z_{P}zP at P P PPP and define P P PPP to be a critical point of f f fff if
(2.2) d f d z P ( P ) = 0 (2.2) d f d z P ( P ) = 0 {:(2.2)(df)/(dz_(P))(P)=0:}\begin{equation*} \frac{d f}{d z_{P}}(P)=0 \tag{2.2} \end{equation*}(2.2)dfdzP(P)=0
The vanishing condition (2.2) is independent of the choice of z P z P z_(P)z_{P}zP, and counted with appropriate multiplicities, the map f f fff has 2 deg ( f ) 2 2 deg ⁡ ( f ) − 2 2deg(f)-22 \operatorname{deg}(f)-22deg⁡(f)−2 critical points. 4 4 ^(4){ }^{4}4
2 To avoid complications, we always work in a subcategory of the category of sets, i.e., all of our objects are sets.
3 3 3quad3 \quad3 More generally, let F = { f 1 , , f r } F = f 1 , … , f r F={f_(1),dots,f_(r)}\mathscr{F}=\left\{f_{1}, \ldots, f_{r}\right\}F={f1,…,fr} be a set of endomorphisms of X X XXX, and let F ⟨ F ⟩ (:F:)\langle\mathscr{F}\rangle⟨F⟩ be the semigroup of maps generated by arbitrary composition of elements of F F F\mathscr{F}F. Then the F F F\mathscr{F}F-orbit of x x xxx is the set O F ( x ) = { f ( x ) : f F } O F ( x ) = { f ( x ) : f ∈ ⟨ F ⟩ } O_(F)(x)={f(x):f in(:F:)}\mathcal{O}_{\mathscr{F}}(x)=\{f(x): f \in\langle\mathcal{F}\rangle\}OF(x)={f(x):f∈⟨F⟩}.
4 More precisely, this is true as long as f f fff is separable, so in particular it is always true in characteristic 0 .
Arithmetic Geometry
rational and integral
points on varieties
rational and integral points on varieties| rational and integral | | :--- | | points on varieties |
rational and integral
points in orbits
rational and integral points in orbits| rational and integral | | :--- | | points in orbits |
torsion points on
abelian varieties
torsion points on abelian varieties| torsion points on | | :--- | | abelian varieties |
periodic and preperiodic
points of rational maps
periodic and preperiodic points of rational maps| periodic and preperiodic | | :--- | | points of rational maps |
abelian varieties with
complex multiplication
abelian varieties with complex multiplication| abelian varieties with | | :--- | | complex multiplication |
postcritically finite
rational maps
postcritically finite rational maps| postcritically finite | | :--- | | rational maps |
"rational and integral points on varieties" "rational and integral points in orbits" "torsion points on abelian varieties" "periodic and preperiodic points of rational maps" "abelian varieties with complex multiplication" "postcritically finite rational maps"| rational and integral <br> points on varieties | rational and integral <br> points in orbits | | :---: | :---: | | torsion points on <br> abelian varieties | periodic and preperiodic <br> points of rational maps | | abelian varieties with <br> complex multiplication | postcritically finite <br> rational maps |
TABLE 1
A dictionary for Arithmetic Dynamics [82, §6.5]
The critical points of an endomorphism f f fff of P 1 P 1 P^(1)\mathbb{P}^{1}P1 are the points at which f f fff fails to be locally bijective. Their location crucially affects the dynamics of f f fff.
Definition 2.1. A (separable) endomorphism f : P 1 P 1 f : P 1 → P 1 f:P^(1)rarrP^(1)f: \mathbb{P}^{1} \rightarrow \mathbb{P}^{1}f:P1→P1 is postcritically finite (PCF) if all of its critical points are preperiodic. PCF maps play a key role in the study of dynamics on P 1 P 1 P^(1)\mathbb{P}^{1}P1.

3. A DICTIONARY FOR ARITHMETIC DYNAMICS

Table 1 gives three fundamental analogies that are used to travel between the worlds of arithmetic geometry and dynamical systems. The associations described in the first two lines of Table 1 are fairly tight, in the sense that they may be used to reformulate many standard results and conjectures in arithmetic geometry as dynamical statements. The following two examples illustrate these connections.
Example 3.1. Let A A AAA be an abelian group, let P A P ∈ A P in AP \in AP∈A, and let f P : A A f P : A → A f_(P):A rarr Af_{P}: A \rightarrow AfP:A→A be the translationby- P P PPP map, i.e., f P ( Q ) = Q + P f P ( Q ) = Q + P f_(P)(Q)=Q+Pf_{P}(Q)=Q+PfP(Q)=Q+P. Then the subgroup of A A AAA generated by P P PPP is the union of two orbits
Z P = O f P ( 0 ) O f P ( 0 ) Z P = O f P ( 0 ) ∪ O f − P ( 0 ) ZP=O_(f_(P))(0)uuO_(f_(-P))(0)\mathbb{Z} P=\mathcal{O}_{f_{P}}(0) \cup \mathcal{O}_{f_{-P}}(0)ZP=OfP(0)∪Of−P(0)
More generally, for any finite set of elements P 1 , , P r A P 1 , … , P r ∈ A P_(1),dots,P_(r)in AP_{1}, \ldots, P_{r} \in AP1,…,Pr∈A, we let P = { ± P 1 , , ± P r } P = ± P 1 , … , ± P r P={+-P_(1),dots,+-P_(r)}\mathscr{P}=\left\{ \pm P_{1}, \ldots, \pm P_{r}\right\}P={±P1,…,±Pr}, and then the subgroup P ⟨ P ⟩ (:P:)\langle\mathcal{P}\rangle⟨P⟩ generated by P 1 , , P r P 1 , … , P r P_(1),dots,P_(r)P_{1}, \ldots, P_{r}P1,…,Pr is the generalized orbit
P = O P ( 0 ) = { f P ( 0 ) : P P } ⟨ P ⟩ = O P ( 0 ) = f P ( 0 ) : P ∈ ⟨ P ⟩ (:P:)=O_(P)(0)={f_(P)(0):P in(:P:)}\langle\mathcal{P}\rangle=\mathcal{O}_{\mathcal{P}}(0)=\left\{f_{P}(0): P \in\langle\mathcal{P}\rangle\right\}⟨P⟩=OP(0)={fP(0):P∈⟨P⟩}
In this way, statements about finitely generated subgroups of abelian varieties may be reformulated as statements about orbits.
Example 3.2. Let G G GGG be a group, let d 2 d ≥ 2 d >= 2d \geq 2d≥2, and let f d : G G f d : G → G f_(d):G rarr Gf_{d}: G \rightarrow Gfd:G→G be the d d ddd-power map f d ( g ) = g d f d ( g ) = g d f_(d)(g)=g^(d)f_{d}(g)=g^{d}fd(g)=gd. Then it is an easy exercise to check that
PrePer ( f ) = G tors PrePer ⁡ ( f ) = G tors  PrePer(f)=G_("tors ")\operatorname{PrePer}(f)=G_{\text {tors }}PrePer⁡(f)=Gtors 
i.e., the elements of G G GGG that are preperiodic for the d d ddd-power map are exactly the elements of G G GGG having finite order. In this way statements about torsion points on abelian varieties may be reformulated as statements about preperiodic points for the multiplication-by- d d ddd map.
Remark 3.3. Examples 3.1 and 3.2 help to justify the associations described in the first two lines of Table 1. The third line is a bit more nebulous. It is a rough analogy based on the following reasoning: 5
  • The CM points in the moduli space A g A g A_(g)\mathcal{A}_{g}Ag of abelian varieties of dimension g g ggg are associated to abelian varieties that have a special algebraic property, namely their endomorphism ring is unusually large. The set of CM points is a countable, Zariski-dense set of points in A g A g A_(g)\mathscr{A}_{g}Ag whose coordinates are algebraic numbers.
  • The PCF points in the moduli space M d 1 M d 1 M_(d)^(1)\mathcal{M}_{d}^{1}Md1 of endomorphisms of P 1 P 1 P^(1)\mathbb{P}^{1}P1 are associated to maps that have a special dynamical property, namely the orbits of their critical points are unusually small. The set of PCF points is a countable, Zariski-dense set of points in M d 1 M d 1 M_(d)^(1)\mathcal{M}_{d}^{1}Md1 whose coordinates are algebraic numbers.
Section 6 describes some progress that helps to justify the third analogy in Table 1. But we must also note that the analogy is not perfect. In particular, CM abelian varieties are abundant in all dimensions, i.e., C M C M CM\mathrm{CM}CM points are Zariski-dense in A g A g A_(g)A_{g}Ag for all g 1 g ≥ 1 g >= 1g \geq 1g≥1. However, evidence suggests that for N 2 N ≥ 2 N >= 2N \geq 2N≥2, PCF maps are not Zariski dense in the moduli space M d N M d N M_(d)^(N)\mathcal{M}_{d}^{N}MdN of endomorphisms of P N P N P^(N)\mathbb{P}^{N}PN; cf. [34].

4. TOPIC #1: DYNAMICAL UNIFORM BOUNDEDNESS

The prototype and motivation for the dynamical uniform boundedness conjecture is the following famous theorem.
Theorem 4.1 ([54]). Let E / Q E / Q E//QE / \mathbb{Q}E/Q be an elliptic curve defined over Q Q Q\mathbb{Q}Q. Then
# E ( Q ) tors 16 # E ( Q ) tors  ≤ 16 #E(Q)_("tors ") <= 16\# E(\mathbb{Q})_{\text {tors }} \leq 16#E(Q)tors ≤16
Remark 4.2. Mazur's theorem was generalized by Kamienny [37] to number fields of small degree, and then by Merel [58], who proved that for all number fields K / Q K / Q K//QK / \mathbb{Q}K/Q and for all elliptic curves E / K E / K E//KE / KE/K, there is a uniform bound
# E ( K ) tors C , where C depends only on the degree [ K : Q ] # E ( K ) tors  ≤ C ,  where  C  depends only on the degree  [ K : Q ] #E(K)_("tors ") <= C,quad" where "C" depends only on the degree "[K:Q]\# E(K)_{\text {tors }} \leq C, \quad \text { where } C \text { depends only on the degree }[K: \mathbb{Q}]#E(K)tors ≤C, where C depends only on the degree [K:Q]
A long-standing conjecture says that the same should be true for abelian varieties A / K A / K A//KA / KA/K of any dimension, where the upper bound depends on [ K : Q ] [ K : Q ] [K:Q][K: \mathbb{Q}][K:Q] and dim ( A ) dim ⁡ ( A ) dim(A)\operatorname{dim}(A)dim⁡(A).
Using the dictionary in Table 1, the theorems of Mazur-Kamienny-Merel and the conjectural abelian variety generalization lead us to a major motivating problem in arithmetic dynamics.
Conjecture 4.3 (Dynamical uniform boundedness conjecture, [62]). Fix integers N 1 N ≥ 1 N >= 1N \geq 1N≥1, d 2 d ≥ 2 d >= 2d \geq 2d≥2, and D 1 D ≥ 1 D >= 1D \geq 1D≥1. There is a constant C ( N , d , D ) C ( N , d , D ) C(N,d,D)C(N, d, D)C(N,d,D) such that for all degree-d morphisms f : P N P N f : P N → P N f:P^(N)rarrP^(N)f: \mathbb{P}^{N} \rightarrow \mathbb{P}^{N}f:PN→PN defined over a number field K K KKK of degree [ K : Q ] = D [ K : Q ] = D [K:Q]=D[K: \mathbb{Q}]=D[K:Q]=D, the number of K K KKK-rational preperiodic points is uniformly bounded,
# PrePer ( f , P N ( K ) ) C ( N , d , D ) # PrePer ⁡ f , P N ( K ) ≤ C ( N , d , D ) #PrePer(f,P^(N)(K)) <= C(N,d,D)\# \operatorname{PrePer}\left(f, \mathbb{P}^{N}(K)\right) \leq C(N, d, D)#PrePer⁡(f,PN(K))≤C(N,d,D)
Remark 4.4. See also [79] for an earlier dynamical uniform boundedness conjecture for K 3 K 3 K3\mathrm{K} 3K3 surfaces admitting noncommuting involutions.
Remark 4.5. Although Conjecture 4.3 only deals with preperiodic points in projective space, it can be used to prove the uniform boundedness conjecture for abelian varieties alluded to in Remark 4.2 [21].
Remark 4.6. Conjecture 4.3 has been generalized to cover quite general families of dynamical systems; see [72, QUESTION 3.2].
Conjecture 4.3 seems out of reach at present. Indeed, even quite special cases present challenges that have not been overcome. We briefly summarize what is known and conjectured in the simplest nontrivial case, which is quadratic polynomials over Q Q Q\mathbb{Q}Q.
Theorem/Conjecture 4.7. For c Q c ∈ Q c inQc \in \mathbb{Q}c∈Q, let f c ( x ) = x 2 + c f c ( x ) = x 2 + c f_(c)(x)=x^(2)+cf_{c}(x)=x^{2}+cfc(x)=x2+c.
(a) Theorem. For each n { 1 , 2 , 3 } n ∈ { 1 , 2 , 3 } n in{1,2,3}n \in\{1,2,3\}n∈{1,2,3}, there are infinitely many c Q c ∈ Q c inQc \in \mathbb{Q}c∈Q such that f c ( x ) f c ( x ) f_(c)(x)f_{c}(x)fc(x) has a Q Q Q\mathbb{Q}Q-rational point of period n n nnn.
(b) Theorem. For all c Q c ∈ Q c inQc \in \mathbb{Q}c∈Q, the polynomial f c ( x ) f c ( x ) f_(c)(x)f_{c}(x)fc(x) does not have a Q Q Q\mathbb{Q}Q-rational point ...
  • of order 4 [ 60 ] ; 4 [ 60 ] ; 4[60];4[60] ;4[60];
  • of order 5 [ 25 ] 5 [ 25 ] 5[25]5[25]5[25];
  • of order 6, conditional on the Birch-Swinnerton-Dyer conjecture [86].
(c) Conjecture. For all n 4 n ≥ 4 n >= 4n \geq 4n≥4, the polynomial f c ( x ) f c ( x ) f_(c)(x)f_{c}(x)fc(x) does not have a Q Q Q\mathbb{Q}Q-rational point of period n n nnn; see [ 91 ] 6 [ 91 ] 6 [91]^(6)[91]^{6}[91]6 and [ 25 ] [ 25 ] [25][25][25].
Remark 4.8. Just as there are elliptic modular curves X 1 ell ( n ) X 1 ell  ( n ) X_(1)^("ell ")(n)X_{1}^{\text {ell }}(n)X1ell (n) whose points classify pairs ( E , P ) ( E , P ) (E,P)(E, P)(E,P) consisting of an elliptic curve E E EEE and an n n nnn-torsion point P P PPP, there are so-called dynatomic modular curves X 1 d y n ( n ) X 1 d y n ( n ) X_(1)^(dyn)(n)X_{1}^{\mathrm{dyn}}(n)X1dyn(n) whose points classify pairs ( c , α ) ( c , α ) (c,alpha)(c, \alpha)(c,α) such that α α alpha\alphaα is a point of period n n nnn for the polynomial f c ( x ) = x 2 + c f c ( x ) = x 2 + c f_(c)(x)=x^(2)+cf_{c}(x)=x^{2}+cfc(x)=x2+c. Mazur's method for proving Theorem 4.1 is to show that X 1 ell ( n ) X 1 ell  ( n ) X_(1)^("ell ")(n)X_{1}^{\text {ell }}(n)X1ell (n) has no (noncuspidal) Q Q Q\mathbb{Q}Q-rational points by mapping X 1 ell ( n ) X 1 ell  ( n ) X_(1)^("ell ")(n)X_{1}^{\text {ell }}(n)X1ell (n) into a carefully chosen quotient A A AAA of its Jacobian variety and showing that the group A ( Q ) A ( Q ) A(Q)A(\mathbb{Q})A(Q) is finite. The proof of Theorem 4.7 starts similarly using X 1 dyn ( n ) X 1 dyn  ( n ) X_(1)^("dyn ")(n)X_{1}^{\text {dyn }}(n)X1dyn (n) instead of X 1 ell ( n ) X 1 ell  ( n ) X_(1)^("ell ")(n)X_{1}^{\text {ell }}(n)X1ell (n), but in this situation, the Jacobian generally does not have a quotient whose group of rational points is finite. Current methods, such as Chabauty-Coleman, for explicitly determining the rational points on curves of high genus (barely) suffice to handle X 1 dyn ( n ) X 1 dyn  ( n ) X_(1)^("dyn ")(n)X_{1}^{\text {dyn }}(n)X1dyn (n) for n 6 n ≤ 6 n <= 6n \leq 6n≤6. The difficulty, or more concretely the difference, between the elliptic curve and dynamical settings centers around the lack of a theory of Hecke correspondences in the dynamical case. (Mea culpa: This simplified explanation is not entirely accurate, but it is meant to convey the overall strategy of the proofs.)
Remark 4.9. Contingent on an appropriate version of the a b c d a b c d abcda b c dabcd-conjecture, the uniform boundedness conjecture has been proven for the family of polynomials x d + c x d + c x^(d)+cx^{d}+cxd+c [47], and more recently for all polynomials [46]. An alternative proof, also using the a b c a b c abca b cabc-conjecture and only valid over Q Q Q\mathbb{Q}Q, says that if d d ddd is sufficiently large and c 1 c ≠ − 1 c!=-1c \neq-1c≠−1, then x d + c x d + c x^(d)+cx^{d}+cxd+c has no Q Q Q\mathbb{Q}Q-rational periodic points other than fixed points [68]. 7 7 ^(7){ }^{7}7
Remark 4.10. A function field analogue of the uniform boundedness conjecture for x d + c x d + c x^(d)+cx^{d}+cxd+c is proven in [ 17 , 18 ] [ 17 , 18 ] [17,18][17,18][17,18]. In the function field setting, the uniformity in the degree [ K : Q ] [ K : Q ] [K:Q][K: \mathbb{Q}][K:Q] described in Conjecture 4.3 is replaced by a bound that depends on the gonality 8 8 ^(8){ }^{8}8 of the field extension.

5. TOPIC #2: DYNAMICAL MODULI SPACES

We fix a field K K KKK and consider parameter and moduli spaces for the set of rational self-maps of P K N P K N P_(K)^(N)\mathbb{P}_{K}^{N}PKN. A rational map f : P K N P K N f : P K N → P K N f:P_(K)^(N)rarrP_(K)^(N)f: \mathbb{P}_{K}^{N} \rightarrow \mathbb{P}_{K}^{N}f:PKN→PKN of degree- d d ddd is specified by an ( N + 1 ) ( N + 1 ) (N+1)(N+1)(N+1) tuple of degree- d d ddd homogeneous polynomials,
f = [ f 0 , , f N ] , f 0 , , f N K [ X 0 , , X n ] f = f 0 , … , f N , f 0 , … , f N ∈ K X 0 , … , X n f=[f_(0),dots,f_(N)],quadf_(0),dots,f_(N)in K[X_(0),dots,X_(n)]f=\left[f_{0}, \ldots, f_{N}\right], \quad f_{0}, \ldots, f_{N} \in K\left[X_{0}, \ldots, X_{n}\right]f=[f0,…,fN],f0,…,fN∈K[X0,…,Xn]
such that f 0 , , f N f 0 , … , f N f_(0),dots,f_(N)f_{0}, \ldots, f_{N}f0,…,fN have no common factors. The map f f fff is a morphism if f 0 , , f N f 0 , … , f N f_(0),dots,f_(N)f_{0}, \ldots, f_{N}f0,…,fN have no common roots in P N ( K ¯ ) P N ( K ¯ ) P^(N)( bar(K))\mathbb{P}^{N}(\bar{K})PN(K¯). We label the coefficients of f 0 , , f N f 0 , … , f N f_(0),dots,f_(N)f_{0}, \ldots, f_{N}f0,…,fN in some specified order as a 1 ( f ) , a 2 ( f ) , , a v ( f ) a 1 ( f ) , a 2 ( f ) , … , a v ( f ) a_(1)(f),a_(2)(f),dots,a_(v)(f)a_{1}(f), a_{2}(f), \ldots, a_{v}(f)a1(f),a2(f),…,av(f), where v = v ( N , d ) := ( N + d d ) ( N + 1 ) v = v ( N , d ) := ( N + d d ) ( N + 1 ) v=v(N,d):=((N+d)/(d))(N+1)v=v(N, d):=\binom{N+d}{d}(N+1)v=v(N,d):=(N+dd)(N+1). Then each such f f fff determines a point
f = [ a 1 ( f ) , , a ν ( f ) ] P ν 1 f = a 1 ( f ) , … , a ν ( f ) ∈ P ν − 1 f=[a_(1)(f),dots,a_(nu)(f)]inP^(nu-1)f=\left[a_{1}(f), \ldots, a_{\nu}(f)\right] \in \mathbb{P}^{\nu-1}f=[a1(f),…,aν(f)]∈Pν−1
There is a homogeneous polynomial R Z [ a 1 , , a ν ] R ∈ Z a 1 , … , a ν RinZ[a_(1),dots,a_(nu)]\mathcal{R} \in \mathbb{Z}\left[a_{1}, \ldots, a_{\nu}\right]R∈Z[a1,…,aν] called the Macaulay resultant having the property that
f = [ f 0 , , f N ] is a morphism R ( a 1 ( f ) , , a v ( f ) ) 0 f = f 0 , … , f N  is a morphism  ⟺ R a 1 ( f ) , … , a v ( f ) ≠ 0 f=[f_(0),dots,f_(N)]" is a morphism "LongleftrightarrowR(a_(1)(f),dots,a_(v)(f))!=0f=\left[f_{0}, \ldots, f_{N}\right] \text { is a morphism } \Longleftrightarrow \mathcal{R}\left(a_{1}(f), \ldots, a_{v}(f)\right) \neq 0f=[f0,…,fN] is a morphism ⟺R(a1(f),…,av(f))≠0
The parameter space of degree- d d ddd endomorphisms of P N P N P^(N)\mathbb{P}^{N}PN is
End d N = { f P ν 1 : R ( f ) 0 } End d N = f ∈ P ν − 1 : R ( f ) ≠ 0 End_(d)^(N)={f inP^(nu-1):R(f)!=0}\operatorname{End}_{d}^{N}=\left\{f \in \mathbb{P}^{\nu-1}: \mathcal{R}(f) \neq 0\right\}EnddN={f∈Pν−1:R(f)≠0}
The isomorphism class of dynamical systems associated to f f fff is the set of all conjugates, i.e., the set of all
f φ = φ 1 f φ , where φ Aut ( P N ) = PGL N + 1 f φ = φ − 1 ∘ f ∘ φ ,  where  φ ∈ Aut ⁡ P N = PGL N + 1 f^(varphi)=varphi^(-1)@f@varphi,quad" where "varphi in Aut(P^(N))=PGL_(N+1)f^{\varphi}=\varphi^{-1} \circ f \circ \varphi, \quad \text { where } \varphi \in \operatorname{Aut}\left(\mathbb{P}^{N}\right)=\operatorname{PGL}_{N+1}fφ=φ−1∘f∘φ, where φ∈Aut⁡(PN)=PGLN+1
Conjugation gives an algebraic action of P G L N + 1 P G L N + 1 PGL_(N+1)\mathrm{PGL}_{N+1}PGLN+1 on the parameter space End d N End d N End_(d)^(N)\operatorname{End}_{d}^{N}EnddN via
(5.1) PGL N + 1 × End d N End d N , ( φ , f ) f φ , (5.1) PGL N + 1 × End d N → End d N , ( φ , f ) ↦ f φ , {:(5.1)PGL_(N+1)xxEnd_(d)^(N)rarrEnd_(d)^(N)","quad(varphi","f)|->f^(varphi)",":}\begin{equation*} \operatorname{PGL}_{N+1} \times \operatorname{End}_{d}^{N} \rightarrow \operatorname{End}_{d}^{N}, \quad(\varphi, f) \mapsto f^{\varphi}, \tag{5.1} \end{equation*}(5.1)PGLN+1×EnddN→EnddN,(φ,f)↦fφ,
and this action extends naturally to P v P v P^(v)\mathbb{P}^{v}Pv
Definition 5.1. The moduli space of degree-d dynamical systems on P N P N P^(N)\mathbb{P}^{N}PN is the quotient space of End d N End d N End_(d)^(N)\operatorname{End}_{d}^{N}EnddN for the conjugation action (5.1),
(5.2) M d N = End d N / P G L N + 1 (5.2) M d N = End d N ⁡ / P G L N + 1 {:(5.2)M_(d)^(N)=End_(d)^(N)//PGL_(N+1):}\begin{equation*} \mathcal{M}_{d}^{N}=\operatorname{End}_{d}^{N} / \mathrm{PGL}_{N+1} \tag{5.2} \end{equation*}(5.2)MdN=EnddN⁡/PGLN+1
It is natural to ask whether the quotient (5.2) can be given some nice sort of structure. Geometric invariant theory (GIT) [63] provides a powerful tool for studying quotients of a variety (or scheme) X X XXX by an infinite algebraic group G G GGG. GIT says that there are stable and semistable loci X s X s s X X s ⊆ X s s ⊆ X X^(s)subeX^(ss)sube XX^{\mathrm{s}} \subseteq X^{\mathrm{ss}} \subseteq XXs⊆Xss⊆X such that there exist quotient varieties (or schemes) X s / / G X s / / G X^(s)////GX^{\mathrm{s}} / / GXs//G and X ss / / G X ss  / / G X^("ss ")////GX^{\text {ss }} / / GXss //G having many agreeable properties. 9 9 ^(9){ }^{9}9
Theorem 5.2. Let N 1 N ≥ 1 N >= 1N \geq 1N≥1 and d 2 d ≥ 2 d >= 2d \geq 2d≥2.
(a) The quotient space M d N ( C ) = End d N ( C ) / PGL N + 1 ( C ) M d N ( C ) = End d N ⁡ ( C ) / PGL N + 1 ⁡ ( C ) M_(d)^(N)(C)=End_(d)^(N)(C)//PGL_(N+1)(C)\mathcal{M}_{d}^{N}(\mathbb{C})=\operatorname{End}_{d}^{N}(\mathbb{C}) / \operatorname{PGL}_{N+1}(\mathbb{C})MdN(C)=EnddN⁡(C)/PGLN+1⁡(C) has a natural structure as an orbifold over C C C\mathbb{C}C [59].
(b) The quotient space M d N = End d N / P G L N + 1 M d N = End d N ⁡ / P G L N + 1 M_(d)^(N)=End_(d)^(N)//PGL_(N+1)\mathcal{M}_{d}^{N}=\operatorname{End}_{d}^{N} / \mathrm{PGL}_{N+1}MdN=EnddN⁡/PGLN+1 has a natural structure as a GIT quotient scheme over Z Z Z\mathbb{Z}Z; see [80] for N = 1 N = 1 N=1N=1N=1 and [44,69] for N 1 . 10 N ≥ 1 . 10 N >= 1.^(10)N \geq 1 .{ }^{10}N≥1.10
It is clear that M d N M d N M_(d)^(N)\mathcal{M}_{d}^{N}MdN is unirational, i.e., it is rationally finitely covered by a projective space, since End d N End d N End_(d)^(N)\operatorname{End}_{d}^{N}EnddN is itself an open subset of a projective space. A subtler question is whether M d N M d N M_(d)^(N)\mathcal{M}_{d}^{N}MdN is rational.
Theorem 5.3. Let d 2 d ≥ 2 d >= 2d \geq 2d≥2.
(a) There is an isomorphism M 2 1 A 2 M 2 1 ≅ A 2 M_(2)^(1)~=A^(2)\mathcal{M}_{2}^{1} \cong \mathbb{A}^{2}M21≅A2, and the semi-stable GIT compactification of M 2 1 M 2 1 M_(2)^(1)\mathcal{M}_{2}^{1}M21 as the quotient of the semi-stable locus in P 5 P 5 P^(5)\mathbb{P}^{5}P5 is isomorphic to P 2 P 2 P^(2)\mathbb{P}^{2}P2 [59,80].
(b) The space M d 1 M d 1 M_(d)^(1)\mathcal{M}_{d}^{1}Md1 is rational, i.e., there exists a birational map P 2 d 2 M d 1 P 2 d − 2 → M d 1 P^(2d-2)rarrM_(d)^(1)\mathbb{P}^{2 d-2} \rightarrow \mathcal{M}_{d}^{1}P2d−2→Md1 [44].
Question 5.4. Is M d N M d N M_(d)^(N)\mathcal{M}_{d}^{N}MdN rational for all d 2 d ≥ 2 d >= 2d \geq 2d≥2 and all N 1 N ≥ 1 N >= 1N \geq 1N≥1 ?
9 9 9quad9 \quad9 For example, over C C C\mathbb{C}C the stable GIT quotient satisfies ( X s / / G ) ( C ) = X s ( C ) / G ( C ) X s / / G ( C ) = X s ( C ) / G ( C ) (X^(s)////G)(C)=X^(s)(C)//G(C)\left(X^{\mathrm{s}} / / G\right)(\mathbb{C})=X^{\mathrm{s}}(\mathbb{C}) / G(\mathbb{C})(Xs//G)(C)=Xs(C)/G(C), i.e., the geometric points of the stable quotient X s / / G X s / / G X^(s)////GX^{\mathrm{s}} / / GXs//G are the G ( C ) G ( C ) G(C)G(\mathbb{C})G(C)-orbits of the geometric points of X X XXX. And the semistable GIT quotient has the property that ( X s s / / G ) ( C ) X s s / / G ( C ) (X^(ss)////G)(C)\left(X^{\mathrm{ss}} / / G\right)(\mathbb{C})(Xss//G)(C) is proper, i.e., it is compact, so it provides a natural compactification of the stable quotient. of S L N + 1 S L N + 1 SL_(N+1)\mathrm{SL}_{N+1}SLN+1 on P ν P ν P^(nu)\mathbb{P}^{\nu}Pν linearized relative to the line bundle O P ν O P ν O_(P^(nu))\mathcal{O}_{\mathbb{P}^{\nu}}OPν (1), and thus the quotient M d N M d N M_(d)^(N)\mathcal{M}_{d}^{N}MdN exists as a GIT quotient scheme over Z Z Z\mathbb{Z}Z.
Just as is done with the moduli space of abelian varieties, it is advantageous to add level structure to dynamical moduli spaces by specifying maps together with points or cycles of various shapes. We start with the case of a single periodic point, and then consider more complicated level structures.
Definition 5.5. For N 1 , n 1 N ≥ 1 , n ≥ 1 N >= 1,n >= 1N \geq 1, n \geq 1N≥1,n≥1, and d 2 d ≥ 2 d >= 2d \geq 2d≥2, we write
End d N [ n ] = { ( f , P ) End d N × P N : P has exact f -period n } End d N ⁡ [ n ] = ( f , P ) ∈ End d N × P N : P  has exact  f -period  n End_(d)^(N)[n]={(f,P)inEnd_(d)^(N)xxP^(N):P" has exact "f"-period "n}\operatorname{End}_{d}^{N}[n]=\left\{(f, P) \in \operatorname{End}_{d}^{N} \times \mathbb{P}^{N}: P \text { has exact } f \text {-period } n\right\}EnddN⁡[n]={(f,P)∈EnddN×PN:P has exact f-period n}
Thus the points of End d N [ n ] End d N ⁡ [ n ] End_(d)^(N)[n]\operatorname{End}_{d}^{N}[n]EnddN⁡[n] classify maps with a marked point of exact period n n nnn.
More generally, we define a (preperiodic) portrait P P P\mathcal{P}P to be the directed graph of a self-map of a finite set of points. (See Figure 1 for an example of a portrait.) Then for a portait P P P\mathcal{P}P having k k kkk vertices, we let 11 11 ^(11){ }^{11}11
End d N [ P ] = { ( f , P 1 , , P k ) End d N × ( P N ) k : P 1 , , P k are f -preperiodic and f : { P 1 , , P k } { P 1 , , P k } is a model for the portrait P } End d N ⁡ [ P ] = f , P 1 , … , P k ∈ End d N × P N k : P 1 , … , P k  are  f -preperiodic and  f : P 1 , … , P k → P 1 , … , P k  is a model for the portrait  P End_(d)^(N)[P]={(f,P_(1),dots,P_(k))inEnd_(d)^(N)xx(P^(N))^(k):[P_(1)","dots","P_(k)" are "f"-preperiodic and "],[quad f:{P_(1),dots,P_(k)}rarr{P_(1),dots,P_(k)}],[" is a model for the portrait "P]}\operatorname{End}_{d}^{N}[\mathcal{P}]=\left\{\left(f, P_{1}, \ldots, P_{k}\right) \in \operatorname{End}_{d}^{N} \times\left(\mathbb{P}^{N}\right)^{k}: \begin{array}{l}P_{1}, \ldots, P_{k} \text { are } f \text {-preperiodic and } \\ \quad f:\left\{P_{1}, \ldots, P_{k}\right\} \rightarrow\left\{P_{1}, \ldots, P_{k}\right\} \\ \text { is a model for the portrait } \mathscr{P}\end{array}\right\}EnddN⁡[P]={(f,P1,…,Pk)∈EnddN×(PN)k:P1,…,Pk are f-preperiodic and f:{P1,…,Pk}→{P1,…,Pk} is a model for the portrait P}.
There is a natural action of φ PGL N + 1 φ ∈ PGL N + 1 varphi inPGL_(N+1)\varphi \in \operatorname{PGL}_{N+1}φ∈PGLN+1 on End d N [ P ] End d N ⁡ [ P ] End_(d)^(N)[P]\operatorname{End}_{d}^{N}[\mathcal{P}]EnddN⁡[P] given by
( f , P 1 , , P k ) φ = ( f φ , φ 1 ( P 1 ) , , φ 1 ( P k ) ) f , P 1 , … , P k φ = f φ , φ − 1 P 1 , … , φ − 1 P k (f,P_(1),dots,P_(k))^(varphi)=(f^(varphi),varphi^(-1)(P_(1)),dots,varphi^(-1)(P_(k)))\left(f, P_{1}, \ldots, P_{k}\right)^{\varphi}=\left(f^{\varphi}, \varphi^{-1}\left(P_{1}\right), \ldots, \varphi^{-1}\left(P_{k}\right)\right)(f,P1,…,Pk)φ=(fφ,φ−1(P1),…,φ−1(Pk))
We denote the resulting quotient space by
M d N [ P ] = End d N [ P ] / P G L N + 1 M d N [ P ] = End d N ⁡ [ P ] / P G L N + 1 M_(d)^(N)[P]=End_(d)^(N)[P]//PGL_(N+1)\mathcal{M}_{d}^{N}[\mathcal{P}]=\operatorname{End}_{d}^{N}[\mathscr{P}] / \mathrm{PGL}_{N+1}MdN[P]=EnddN⁡[P]/PGLN+1
If C n C n C_(n)\mathscr{C}_{n}Cn is a portrait consisting of a single n n nnn-cycle, then End d N [ C n ] End d N [ n ] End d N ⁡ C n ≅ End d N ⁡ [ n ] End_(d)^(N)[C_(n)]~=End_(d)^(N)[n]\operatorname{End}_{d}^{N}\left[\mathcal{C}_{n}\right] \cong \operatorname{End}_{d}^{N}[n]EnddN⁡[Cn]≅EnddN⁡[n], and we write M d N [ n ] M d N [ n ] M_(d)^(N)[n]\mathcal{M}_{d}^{N}[n]MdN[n] for M d N [ C n ] M d N C n M_(d)^(N)[C_(n)]\mathcal{M}_{d}^{N}\left[\mathscr{C}_{n}\right]MdN[Cn].

FIGURE 1

A portrait consisting of a 3-cycle, a 4-cycle, and three other preperiodic points
Theorem 5.6 ([20]). Let P P P\mathcal{P}P be a preperiodic portrait. 12 12 ^(12){ }^{12}12 Then the quotient space M d N [ P ] M d N [ P ] M_(d)^(N)[P]\mathcal{M}_{d}^{N}[\mathcal{P}]MdN[P] described in (5.3) exists 13 13 ^(13){ }^{13}13 as a GIT geometric quotient scheme over Z Z Z\mathbb{Z}Z.
11 This definition of End d N [ P ] End d N ⁡ [ P ] End_(d)^(N)[P]\operatorname{End}_{d}^{N}[\mathscr{P}]EnddN⁡[P] conveys the right idea; see [20] for a rigorous definition.
12 More generally, one can construct the moduli space M d N [ P ] M d N [ P ] M_(d)^(N)[P]\mathcal{M}_{d}^{N}[\mathcal{P}]MdN[P] associated to a portrait P P P\mathscr{P}P that includes nonpreperiodic points and/or whose vertices are assigned multiplicities.
13 There is a precise combinatorial-geometric characterization of the portraits P P P\mathcal{P}P for which M d 1 [ P ] ( C ) M d 1 [ P ] ( C ) ≠ ∅ M_(d)^(1)[P](C)!=O/\mathcal{M}_{d}^{1}[\mathcal{P}](\mathbb{C}) \neq \emptysetMd1[P](C)≠∅, but analogous characterizations for N 2 N ≥ 2 N >= 2N \geq 2N≥2 and/or in positive characteristic are not currently known.
It is known [87] that the moduli space A g A g A_(g)\mathscr{A}_{g}Ag of principally polarized abelian varieties is of general type for all g 9 g ≥ 9 g >= 9g \geq 9g≥9. Analogous results for dynamical moduli spaces are still unknown, but our dictionary yields some conjectures. 14 14 ^(14){ }^{14}14
Conjecture 5.7. Let N 1 N ≥ 1 N >= 1N \geq 1N≥1 and d 2 d ≥ 2 d >= 2d \geq 2d≥2.
(a) For all n 1 n ≥ 1 n >= 1n \geq 1n≥1, the moduli space M d N [ n ] M d N [ n ] M_(d)^(N)[n]\mathcal{M}_{d}^{N}[n]MdN[n] is irreducible.
(b) For all sufficiently large n n nnn, depending on N N NNN and d d ddd, the moduli space M d N [ n ] M d N [ n ] M_(d)^(N)[n]\mathcal{M}_{d}^{N}[n]MdN[n] is a variety of general type.
Remark 5.8. The moduli space M 2 1 [ n ] M 2 1 [ n ] M_(2)^(1)[n]\mathcal{M}_{2}^{1}[n]M21[n] of degree-2 endomorphisms of P 1 P 1 P^(1)\mathbb{P}^{1}P1 is a finite cover of M 2 1 A 2 M 2 1 ≅ A 2 M_(2)^(1)~=A^(2)\mathcal{M}_{2}^{1} \cong \mathbb{A}^{2}M21≅A2, so it is a surface. It is known to be irreducible for all n 1 n ≥ 1 n >= 1n \geq 1n≥1 [48]. For 1 n 5 1 ≤ n ≤ 5 1 <= n <= 51 \leq n \leq 51≤n≤5, the surface M 2 1 [ n ] M 2 1 [ n ] M_(2)^(1)[n]\mathcal{M}_{2}^{1}[n]M21[n] is a rational surface, while M 2 1 [ 6 ] M 2 1 [ 6 ] M_(2)^(1)[6]\mathcal{M}_{2}^{1}[6]M21[6] is a surface of general type [12].
Remark 5.9. Tai's proof [87] that A g A g A_(g)\mathscr{A}_{g}Ag is of general type relies on the theory of theta functions, which are used to create sections of the canonical bundle. There are similarly naturally defined functions on M d N M d N M_(d)^(N)\mathcal{M}_{d}^{N}MdN, and more generally on M d N [ n ] M d N [ n ] M_(d)^(N)[n]\mathcal{M}_{d}^{N}[n]MdN[n], that are defined using multiplier systems. 15 15 ^(15){ }^{15}15 For N = 1 N = 1 N=1N=1N=1, it is known that a multiplier system of sufficiently high degree gives a map M d 1 A r M d 1 → A r M_(d)^(1)rarrA^(r)\mathcal{M}_{d}^{1} \rightarrow \mathbb{A}^{r}Md1→Ar that is (essentially) finite-to-one 16 16 ^(16){ }^{16}16 onto its image [55]. So although the analogy between theta functions on A g A g A_(g)\mathcal{A}_{g}Ag and multiplier system functions on M d N M d N M_(d)^(N)\mathcal{M}_{d}^{N}MdN is tenuous at best, the latter currently provide one of the most natural ways to create dynamically defined functions on dynamical moduli spaces.
A map f End d N ( K ) f ∈ End d N ⁡ ( K ) f inEnd_(d)^(N)(K)f \in \operatorname{End}_{d}^{N}(K)f∈EnddN⁡(K) defined over K K KKK with a K K KKK-rational n n nnn-periodic point P P N ( K ) P ∈ P N ( K ) P inP^(N)(K)P \in \mathbb{P}^{N}(K)P∈PN(K) gives a K K KKK-rational point f , P M d N [ n ] ( K ) ⟨ f , P ⟩ ∈ M d N [ n ] ( K ) (:f,P:)inM_(d)^(N)[n](K)\langle f, P\rangle \in \mathcal{M}_{d}^{N}[n](K)⟨f,P⟩∈MdN[n](K). The dynamical uniform boundedness conjecture (Conjecture 4.3) is thus closely related to the question of K K KKK-rational points on dynamical moduli spaces. We formulate a uniform boundedness conjecture for such spaces.
Conjecture 5.10 (Dynamical uniform boundedness conjecture: version 2). Fix integers N 1 , d 2 N ≥ 1 , d ≥ 2 N >= 1,d >= 2N \geq 1, d \geq 2N≥1,d≥2, and D 1 D ≥ 1 D >= 1D \geq 1D≥1. There is a constant C ( N , d , D ) C ′ ( N , d , D ) C^(')(N,d,D)C^{\prime}(N, d, D)C′(N,d,D) such that for all number fields K K KKK of degree [ K : Q ] = D [ K : Q ] = D [K:Q]=D[K: \mathbb{Q}]=D[K:Q]=D and all preperiodic portraits P P P\mathcal{P}P,
( # { vertices of P } C ( N , d , D ) ) M d N [ P ] ( K ) = # {  vertices of  P } ≥ C ′ ( N , d , D ) ⟹ M d N [ P ] ( K ) = ∅ (#{" vertices of "P} >= C^(')(N,d,D))LongrightarrowM_(d)^(N)[P](K)=O/\left(\#\{\text { vertices of } \mathcal{P}\} \geq C^{\prime}(N, d, D)\right) \Longrightarrow \mathcal{M}_{d}^{N}[\mathcal{P}](K)=\emptyset(#{ vertices of P}≥C′(N,d,D))⟹MdN[P](K)=∅
gonality.
Briefly, for N = 1 N = 1 N=1N=1N=1, let k 1 k ≥ 1 k >= 1k \geq 1k≥1, let f End d 1 f ∈ End d 1 f inEnd_(d)^(1)f \in \operatorname{End}_{d}^{1}f∈Endd1, and let P 1 , , P r P 1 , … , P r P_(1),dots,P_(r)P_{1}, \ldots, P_{r}P1,…,Pr be the periodic points of f f fff with period dividing k k kkk. The derivatives ( f k ) ( P i ) f k ′ P i (f^(k))^(')(P_(i))\left(f^{k}\right)^{\prime}\left(P_{i}\right)(fk)′(Pi) are P G L 2 P G L 2 PGL_(2)\mathrm{PGL}_{2}PGL2-conjugate independent, and the k k kkk-level multiplier system of f f fff is the list Λ k ( f ) Λ k ( f ) Lambda_(k)(f)\Lambda_{k}(f)Λk(f) of the elementary symmetric functions of ( f k ) ( P 1 ) , , ( f k ) ( P r ) f k ′ P 1 , … , f k ′ P r (f^(k))^(')(P_(1)),dots,(f^(k))^(')(P_(r))\left(f^{k}\right)^{\prime}\left(P_{1}\right), \ldots,\left(f^{k}\right)^{\prime}\left(P_{r}\right)(fk)′(P1),…,(fk)′(Pr). Then Λ k ( f ) Λ k ( f ) Lambda_(k)(f)\Lambda_{k}(f)Λk(f) gives a well-defined morphism Λ k ( f ) Λ k ( f ) Lambda_(k)(f)\Lambda_{k}(f)Λk(f) : M d 1 A r M d 1 → A r M_(d)^(1)rarrA^(r)\mathcal{M}_{d}^{1} \rightarrow \mathbb{A}^{r}Md1→Ar.
16 More precisely, the map is finite-to-one unless n n nnn is a square, in which case it maps the j j jjj line of flexible Lattès maps to a single point. This is thus one of those results that's "true except in the obvious cases where it is false."
Remark 5.11. It is clear that Conjecture 5.10 implies Conjecture 4.3. The opposite implication is also true, but the proof is more difficult due to the Field-of-Moduli versus Field-ofDefinition Problem. The key step, proven in [19] and [20, SEctions 16-17], is to show that every point in M d N [ P ] ( K ) M d N [ P ] ( K ) M_(d)^(N)[P](K)\mathcal{M}_{d}^{N}[\mathcal{P}](K)MdN[P](K) is represented by a point in End d N [ P ] ( L ) End d N ⁡ [ P ] ( L ) End_(d)^(N)[P](L)\operatorname{End}_{d}^{N}[\mathscr{P}](L)EnddN⁡[P](L) defined over an extension L / K L / K L//KL / KL/K whose degree is bounded solely by d d ddd and N N NNN. When N = 1 N = 1 N=1N=1N=1, one can take [ L : K ] 2 [ L : K ] ≤ 2 [L:K] <= 2[L: K] \leq 2[L:K]≤2 [32], but for N 2 N ≥ 2 N >= 2N \geq 2N≥2 it is an open question whether [ L : K ] [ L : K ] [L:K][L: K][L:K] needs to depend on d d ddd.
Within M d N M d N M_(d)^(N)\mathcal{M}_{d}^{N}MdN and its GIT semistable compactification M ¯ d N M ¯ d N bar(M)_(d)^(N)\overline{\mathcal{M}}_{d}^{N}M¯dN lie many interesting subvarieties. For example:
  • The space of polynomial maps 17 17 ^(17){ }^{17}17
Poly d N = { f M d N : f comes from a morphism A N A N } Poly d N = f ∈ M d N : f  comes from a morphism  A N → A N Poly_(d)^(N)={f inM_(d)^(N):f" comes from a morphism "A^(N)rarrA^(N)}\operatorname{Poly}_{d}^{N}=\left\{f \in \mathcal{M}_{d}^{N}: f \text { comes from a morphism } \mathbb{A}^{N} \rightarrow \mathbb{A}^{N}\right\}PolydN={f∈MdN:f comes from a morphism AN→AN}
is a subvariety of M d N M d N M_(d)^(N)\mathcal{M}_{d}^{N}MdN satisfying dim ( Poly d 1 ) = N N + 1 dim ( M d N ) dim ⁡ Poly ⁡ d 1 = N N + 1 dim ⁡ M d N dim(Poly _(d)^(1))=(N)/(N+1)dim(M_(d)^(N))\operatorname{dim}\left(\operatorname{Poly}{ }_{d}^{1}\right)=\frac{N}{N+1} \operatorname{dim}\left(\mathcal{M}_{d}^{N}\right)dim⁡(Poly⁡d1)=NN+1dim⁡(MdN).
  • Iteration of dominant rational maps presents its own interesting challenges; see Section 8 for some examples. The set of degree d d ddd dominant rational maps P N P N P N → P N P^(N)rarrP^(N)\mathbb{P}^{N} \rightarrow \mathbb{P}^{N}PN→PN is a Zariski open subvariety of P ν 1 ( C ) P ν − 1 ( C ) P^(nu-1)(C)\mathbb{P}^{\nu-1}(\mathbb{C})Pν−1(C) [81, PROPOSItion 7], but the locus of points in ( M ¯ d N M d N ) ( C ) M ¯ d N ∖ M d N ( C ) ( bar(M)_(d)^(N)\\M_(d)^(N))(C)\left(\overline{\mathcal{M}}_{d}^{N} \backslash \mathcal{M}_{d}^{N}\right)(\mathbb{C})(M¯dN∖MdN)(C) arising from dominant rational maps is not well understood; cf. [42].
The spaces of polynomial maps and dominant rational maps have large dimension. At the other extreme are various 1-parameter families of maps that have been much studied, starting with the ubiquitous family of quadratic polynomials
f c ( x ) = x 2 + c f c ( x ) = x 2 + c f_(c)(x)=x^(2)+cf_{c}(x)=x^{2}+cfc(x)=x2+c
that gives a line A 1 A 1 A^(1)\mathbb{A}^{1}A1 in M 2 1 A 2 M 2 1 ≅ A 2 M_(2)^(1)~=A^(2)\mathcal{M}_{2}^{1} \cong \mathbb{A}^{2}M21≅A2. Adding level structure leads to a dynamical analogue of the classical elliptic modular curve X 1 ell ( n ) X 1 ell  ( n ) X_(1)^("ell ")(n)X_{1}^{\text {ell }}(n)X1ell (n) that classifies pairs ( E , P ) ( E , P ) (E,P)(E, P)(E,P) consisting of an elliptic curve E E EEE and an n n nnn-torsion point P P PPP. In the dynamical setting, we replace the n n nnn-torsion point with a point of period n n nnn, but the following example shows that some care is needed.
Example 5.12. The polynomial f ( x ) = x 2 3 4 f ( x ) = x 2 − 3 4 f(x)=x^(2)-(3)/(4)f(x)=x^{2}-\frac{3}{4}f(x)=x2−34 has no points of exact period 2, since
f ( x ) x = ( 2 x + 1 ) ( 2 x 3 ) and f 2 ( x ) x = ( 2 x + 1 ) 3 ( 2 x 3 ) f ( x ) − x = ( 2 x + 1 ) ( 2 x − 3 )  and  f 2 ( x ) − x = ( 2 x + 1 ) 3 ( 2 x − 3 ) f(x)-x=(2x+1)(2x-3)quad" and "quadf^(2)(x)-x=(2x+1)^(3)(2x-3)f(x)-x=(2 x+1)(2 x-3) \quad \text { and } \quad f^{2}(x)-x=(2 x+1)^{3}(2 x-3)f(x)−x=(2x+1)(2x−3) and f2(x)−x=(2x+1)3(2x−3)
But since f 2 ( x ) x f ( x ) x = ( 2 x + 1 ) 2 f 2 ( x ) − x f ( x ) − x = ( 2 x + 1 ) 2 (f^(2)(x)-x)/(f(x)-x)=(2x+1)^(2)\frac{f^{2}(x)-x}{f(x)-x}=(2 x+1)^{2}f2(x)−xf(x)−x=(2x+1)2, we say that x = 1 2 x = − 1 2 x=-(1)/(2)x=-\frac{1}{2}x=−12 is a point of formal period 2 for f ( x ) . 18 f ( x ) . 18 f(x).^(18)f(x) .^{18}f(x).18
17 For example, the space Poly d 1 M d 1 d 1 ⊂ M d 1 _(d)^(1)subM_(d)^(1){ }_{d}^{1} \subset \mathcal{M}_{d}^{1}d1⊂Md1 is the space of polynomials x d + a 2 x d 2 + + a d x d + a 2 x d − 2 + ⋯ + a d x^(d)+a_(2)x^(d-2)+cdots+a_(d)x^{d}+a_{2} x^{d-2}+\cdots+a_{d}xd+a2xd−2+⋯+ad modulo the conjugation x ζ x x → ζ x x rarr zeta xx \rightarrow \zeta xx→ζx for a primitive ( d 1 ) ( d − 1 ) (d-1)(d-1)(d−1)-root of unity ζ ζ zeta\zetaζ, so Poly d 1 d 1 _(d)^(1){ }_{d}^{1}d1 is a quotient of A d 1 A d − 1 A^(d-1)\mathbb{A}^{d-1}Ad−1 by a finite group.
In general, points of formal period n n nnn for the polynomial f ( x ) f ( x ) f(x)f(x)f(x) are roots of the dynatomic polynomial
Φ f ( x ) := d n ( f d ( x ) x ) μ ( n / d ) Φ f ( x ) := ∏ d ∣ n   f d ( x ) − x μ ( n / d ) Phi_(f)(x):=prod_(d∣n)(f^(d)(x)-x)^(mu(n//d))\Phi_{f}(x):=\prod_{d \mid n}\left(f^{d}(x)-x\right)^{\mu(n / d)}Φf(x):=∏d∣n(fd(x)−x)μ(n/d)
where μ μ mu\muμ is the Möbius function. Dynatomic polynomials are thus dynamical analogues of classical cyclotomic polynomials, but with the caveat that Φ f ( x ) Φ f ( x ) Phi_(f)(x)\Phi_{f}(x)Φf(x) may have roots of multiplicity greater than 1 , even in characteristic 0 . In higher dimension, the points of formal period n n nnn give a dynatomic 0 -cycle whose effectivity is proven in [33].
Definition 5.13. The level n n nnn dynatomic curve 19 19 ^(19){ }^{19}19 (associated to x 2 + c x 2 + c x^(2)+cx^{2}+cx2+c ) is the affine curve
Y 1 d y n ( n ) = { ( c , α ) A 2 : α is a point of formal period n for f c ( x ) = x 2 + c } Y 1 d y n ( n ) = ( c , α ) ∈ A 2 : α  is a point of formal period  n  for  f c ( x ) = x 2 + c Y_(1)^(dyn)(n)={(c,alpha)inA^(2):alpha" is a point of formal period "n" for "f_(c)(x)=x^(2)+c}Y_{1}^{\mathrm{dyn}}(n)=\left\{(c, \alpha) \in \mathbb{A}^{2}: \alpha \text { is a point of formal period } n \text { for } f_{c}(x)=x^{2}+c\right\}Y1dyn(n)={(c,α)∈A2:α is a point of formal period n for fc(x)=x2+c}
The desingularized projective completion of Y 1 d y n ( n ) Y 1 d y n ( n ) Y_(1)^(dyn)(n)Y_{1}^{\mathrm{dyn}}(n)Y1dyn(n) is denoted X 1 d y n ( n ) X 1 d y n ( n ) X_(1)^(dyn)(n)X_{1}^{\mathrm{dyn}}(n)X1dyn(n). The points in the complement X 1 d y n ( n ) Y 1 d y n ( n ) X 1 d y n ( n ) ∖ Y 1 d y n ( n ) X_(1)^(dyn)(n)\\Y_(1)^(dyn)(n)X_{1}^{\mathrm{dyn}}(n) \backslash Y_{1}^{\mathrm{dyn}}(n)X1dyn(n)∖Y1dyn(n), which correspond to degenerate maps, are called cusps. 20 20 ^(20){ }^{20}20
Points in Y 1 d y n ( n ) ( K ) Y 1 d y n ( n ) ( K ) Y_(1)^(dyn)(n)(K)Y_{1}^{\mathrm{dyn}}(n)(K)Y1dyn(n)(K) classify quadratic polynomials defined over K K KKK having a K K KKK-rational point of period n n nnn, so a version of Theorem/Conjecture 4.7 says that
Exercise X 1 d y n ( n ) P 1 for n { 1 , 2 , 3 } Theorem X 1 d y n ( n ) ( Q ) = { cusps } for n { 4 , 5 , 6 } , Conjecture X 1 d y n ( n ) ( Q ) = { cusps } for all n 4  Exercise  X 1 d y n ( n ) ≅ P 1  for  n ∈ { 1 , 2 , 3 }  Theorem  X 1 d y n ( n ) ( Q ) = {  cusps  }  for  n ∈ { 4 , 5 , 6 } ,  Conjecture  X 1 d y n ( n ) ( Q ) = {  cusps  }  for all  n ≥ 4 {:[" Exercise ",X_(1)^(dyn)(n)~=P^(1)quad" for "n in{1","2","3}],[" Theorem ",X_(1)^(dyn)(n)(Q)={" cusps "}," for "n in{4","5","6}","],[" Conjecture ",X_(1)^(dyn)(n)(Q)={" cusps "}," for all "n >= 4]:}\begin{array}{rll} \text { Exercise } & X_{1}^{\mathrm{dyn}}(n) \cong \mathbb{P}^{1} \quad \text { for } n \in\{1,2,3\} \\ \text { Theorem } & X_{1}^{\mathrm{dyn}}(n)(\mathbb{Q})=\{\text { cusps }\} & \text { for } n \in\{4,5,6\}, \\ \text { Conjecture } & X_{1}^{\mathrm{dyn}}(n)(\mathbb{Q})=\{\text { cusps }\} & \text { for all } n \geq 4 \end{array} Exercise X1dyn(n)≅P1 for n∈{1,2,3} Theorem X1dyn(n)(Q)={ cusps } for n∈{4,5,6}, Conjecture X1dyn(n)(Q)={ cusps } for all n≥4
Much is known about the geometry of X 1 d y n ( n ) X 1 d y n ( n ) X_(1)^(dyn)(n)X_{1}^{\mathrm{dyn}}(n)X1dyn(n), as summarized in the next result, although we note that even the proof that X 1 d y n ( n ) X 1 d y n ( n ) X_(1)^(dyn)(n)X_{1}^{\mathrm{dyn}}(n)X1dyn(n) is geometrically irreducible relies on dynamical properties of x 2 + c x 2 + c x^(2)+cx^{2}+cx2+c as reflected in the geometry of the Mandelbrot set.
Theorem 5.14. Let X 1 d y n ( n ) X 1 d y n ( n ) X_(1)^(dyn)(n)X_{1}^{\mathrm{dyn}}(n)X1dyn(n) be the smooth projective dynatomic curve associated to x 2 + c x 2 + c x^(2)+cx^{2}+cx2+c.
(a) The dynatomic modular curve X 1 d y n ( n ) X 1 d y n ( n ) X_(1)^(dyn)(n)X_{1}^{\mathrm{dyn}}(n)X1dyn(n) is geometrically irreducible over C [ 13 C [ 13 C[13\mathbb{C}[13C[13, 41 , 76 ] 41 , 76 ] 41,76]41,76]41,76]. 21
(b) There is an explicit, but rather complicated, formula for the genus of X 1 d y n ( n ) X 1 d y n ( n ) X_(1)^(dyn)(n)X_{1}^{\mathrm{dyn}}(n)X1dyn(n) [61]. In any case, genus ( X 1 d y n ( n ) ) X 1 d y n ( n ) → ∞ (X_(1)^(dyn)(n))rarr oo\left(X_{1}^{\mathrm{dyn}}(n)\right) \rightarrow \infty(X1dyn(n))→∞ as n n → ∞ n rarr oon \rightarrow \inftyn→∞.
(c) The gonality 22 22 ^(22){ }^{22}22 of X 1 d y n ( n ) X 1 d y n ( n ) X_(1)^(dyn)(n)X_{1}^{\mathrm{dyn}}(n)X1dyn(n) goes to ∞ oo\infty∞ as n n → ∞ n rarr oon \rightarrow \inftyn→∞ [18].

6. TOPIC #3: DYNAMICAL UNLIKELY INTERSECTIONS

The guiding philosophy of unlikely intersections in arithmetic geometry is the following general, albeit somewhat vague, principle.
19 There are dynatomic curves associated to many other interesting 1-parameter families of maps, including, for example, the family of degree- d d ddd unicritical polynomials f d , c ( x ) = x d + c f d , c ( x ) = x d + c f_(d,c)(x)=x^(d)+cf_{d, c}(x)=x^{d}+cfd,c(x)=xd+c and the family of degree-2 rational maps g b ( x ) = x / ( x 2 + b ) g b ( x ) = x / x 2 + b g_(b)(x)=x//(x^(2)+b)g_{b}(x)=x /\left(x^{2}+b\right)gb(x)=x/(x2+b) that admit a nontrivial automorphism g b ( x ) = g b ( x ) g b ( − x ) = − g b ( x ) g_(b)(-x)=-g_(b)(x)g_{b}(-x)=-g_{b}(x)gb(−x)=−gb(x). We mention that there is a natural action of f f fff on Y 1 d y n ( n ) Y 1 d y n ( n ) Y_(1)^(dyn)(n)Y_{1}^{\mathrm{dyn}}(n)Y1dyn(n) defined by ( c , α ) ( c , f ( α ) ) ( c , α ) ↦ ( c , f ( α ) ) (c,alpha)|->(c,f(alpha))(c, \alpha) \mapsto(c, f(\alpha))(c,α)↦(c,f(α)), and that the quotient curve Y 0 d y n ( n ) = Y 1 d y n ( n ) / f Y 0 d y n ( n ) = Y 1 d y n ( n ) / ⟨ f ⟩ Y_(0)^(dyn)(n)=Y_(1)^(dyn)(n)//(:f:)Y_{0}^{\mathrm{dyn}}(n)=Y_{1}^{\mathrm{dyn}}(n) /\langle f\rangleY0dyn(n)=Y1dyn(n)/⟨f⟩ and its completion X 0 d y n ( n ) X 0 d y n ( n ) X_(0)^(dyn)(n)X_{0}^{\mathrm{dyn}}(n)X0dyn(n) provide analogues of the elliptic modular curve X 0 ell ( n ) X 0 ell  ( n ) X_(0)^("ell ")(n)X_{0}^{\text {ell }}(n)X0ell (n).
21 More generally, the dynatomic modular curves associated to the family of unicritical polynomials x d + c x d + c x^(d)+cx^{d}+cxd+c are irreducible. However, the dynatomic modular curves associated to the family x / ( x 2 + b ) x / x 2 + b x//(x^(2)+b)x /\left(x^{2}+b\right)x/(x2+b) turn out to be reducible for even n n nnn; see [48]. The gonality of an algebraic curve X X XXX is the minimal degree of a nonconstant map X P 1 X → P 1 X rarrP^(1)X \rightarrow \mathbb{P}^{1}X→P1

The Tao of Unlikely Intersections

Let X X XXX be an algebraic variety, let Y X Y ⊆ X Y sube XY \subseteq XY⊆X be an algebraic subvariety of X X XXX, and let Γ X Γ ⊂ X Gamma sub X\Gamma \subset XΓ⊂X be an "interesting" countable subset of X X XXX. Then
Γ Y Γ ∩ Y Gamma nn Y\Gamma \cap YΓ∩Y is sparse (except when it is "obviously" not).
Slightly more precisely, if Γ Y Γ ∩ Y Gamma nn Y\Gamma \cap YΓ∩Y is Zariski dense in Y Y YYY, then there should be a geometric reason that explains its density.
We recall two famous unlikely intersection theorems from arithmetic geometry, which we initially state in an intuitively appealing, though somewhat whimsical, manner.
Theorem 6.1 (Mordell-Lang conjecture, [23,90]). Let A / C A / C A//CA / \mathbb{C}A/C be an abelian variety, let Y A Y ⊆ A Y sube AY \subseteq AY⊆A be a subvariety of A A AAA, and let Γ A ( C ) Γ ⊂ A ( C ) Gamma sub A(C)\Gamma \subset A(\mathbb{C})Γ⊂A(C) be a finitely generated subgroup of A A AAA. Then 23 23 ^(23){ }^{23}23
Γ Y Γ ∩ Y Gamma nn Y\Gamma \cap YΓ∩Y is not Zariski dense in Y Y YYY (except when it "obviously" is).
Theorem 6.2 (Manin-Mumford conjecture, [ 73 , 74 ] [ 73 , 74 ] [73,74][73,74][73,74] ). Let A / C A / C A//CA / \mathbb{C}A/C be an abelian variety, and let Y A Y ⊆ A Y sube AY \subseteq AY⊆A be a subvariety of A A AAA. Then
A tors Y A tors  ∩ Y A_("tors ")nn YA_{\text {tors }} \cap YAtors ∩Y is not Zariski dense in Y Y YYY (except when it "obviously" is).
The actual statements of Theorems 6.1 and 6.2 explain quite precisely that if Y Y YYY is saturated with special points, then there is a geometric reason for that saturation.
Theorem 6.3 (Rigorous formulation of Theorems 6.1 and 6.2). If Γ Y Γ ∩ Y Gamma nn Y\Gamma \cap YΓ∩Y or A tors Y A tors  ∩ Y A_("tors ")nn YA_{\text {tors }} \cap YAtors ∩Y is Zariski dense in Y Y YYY, then Y Y YYY is necessarily a translate of an abelian subvariety of A A AAA by a torsion point of A A AAA.
Remark 6.4. Theorems 6.1 and 6.2 may be combined and strengthened by replacing the abelian variety A A AAA with a semi-abelian variety and by replacing Γ Î“ Gamma\GammaΓ with its divisible subgroup n 0 [ n ] 1 ( Γ ) ⋃ n ≥ 0   [ n ] − 1 ( Γ ) uuu_(n >= 0)[n]^(-1)(Gamma)\bigcup_{n \geq 0}[n]^{-1}(\Gamma)⋃n≥0[n]−1(Γ); see [56].
Theorem 6.1 says that points in a finitely generated subgroup Γ Î“ Gamma\GammaΓ generally do not lie on a subvariety. According to Table 1, for the dynamical analogue of Theorem 6.1 we should replace the group Γ Î“ Gamma\GammaΓ with the points in an orbit. This leads to our first dynamical unlikely intersection conjecture.
Conjecture 6.5 (Dynamical Mordell-Lang conjecture). Let X / C X / C X//CX / \mathbb{C}X/C be a smooth quasiprojective variety, let f : X X f : X → X f:X rarr Xf: X \rightarrow Xf:X→X be a regular self-map of X X XXX, let P X ( C ) P ∈ X ( C ) P in X(C)P \in X(\mathbb{C})P∈X(C) be a point with infinite f f fff-orbit, and let Y X Y ⊆ X Y sube XY \subseteq XY⊆X be a subvariety of X X XXX. Then
O f ( P ) Y O f ( P ) ∩ Y O_(f)(P)nn Y\mathcal{O}_{f}(P) \cap YOf(P)∩Y is not Zariski dense in Y Y YYY (except when it "obviously" is).
Rigorous Formulation #1. If O f ( P ) Y O f ( P ) ∩ Y O_(f)(P)nn Y\mathcal{O}_{f}(P) \cap YOf(P)∩Y is Zariski dense, then Y Y YYY is f f fff-periodic. 24 24 ^(24){ }^{24}24
2 3 2 3 23\mathbf{2 3}23 The proof of Theorem 6.1 uses methods from Diophantine approximation. An earlier proof in the case that Y Y YYY is a curve of genus at least 2 used moduli-theoretic techniques [22].
24 We says that Z Z ZZZ is f f fff-periodic if there is an integer n > 0 n > 0 n > 0n>0n>0 such that f n ( Z ) = Z f n ( Z ) = Z f^(n)(Z)=Zf^{n}(Z)=Zfn(Z)=Z.
Rigorous Formulation #2. The set
{ n 0 : f n ( P ) Y } n ≥ 0 : f n ( P ) ∈ Y {n >= 0:f^(n)(P)in Y}\left\{n \geq 0: f^{n}(P) \in Y\right\}{n≥0:fn(P)∈Y}
is a finite union of one-sided arithmetic progressions [29]. 25 25 ^(25){ }^{25}25
Example 6.6. Among the known cases of the dynamical Mordell-Lang conjecture, we cite the following:
Unramified maps. Conjecture 6.5 is true for étale morphisms of quasiprojective varieties [7]. See the monograph [8] for additional information.
Endomorphisms of A 2 A 2 A^(2)\mathbb{A}^{\mathbf{2}}A2. Conjecture 6.5 is true for all endomorphisms of A 2 A 2 A^(2)\mathbb{A}^{2}A2 defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ [92].
Split endomorphisms. Conjecture 6.5 is true for split endomorphisms of ( P 1 ) n P 1 n (P^(1))^(n)\left(\mathbb{P}^{1}\right)^{n}(P1)n, which are maps of the form f 1 ( P 1 ) × × f n ( P n ) f 1 P 1 × ⋯ × f n P n f_(1)(P_(1))xx cdots xxf_(n)(P_(n))f_{1}\left(P_{1}\right) \times \cdots \times f_{n}\left(P_{n}\right)f1(P1)×⋯×fn(Pn) [11], and more generally for certain skew-split endomorphisms [31].
Remark 6.7. The dynamical Mordell-Lang conjecture has also been investigated in characteristic p p ppp, although the statement may need a tweak. For example, if f f fff is a projective surface automorphism or a birational endomorphism of A 2 A 2 A^(2)\mathbb{A}^{2}A2 whose dynamical degree (see Section 8) satisfies δ f > 1 δ f > 1 delta_(f) > 1\delta_{f}>1δf>1, then Conjecture 6.5 is true in all characteristics [94]. For other results in finite characteristic, see, for example, [ 8 , 14 , 26 ] [ 8 , 14 , 26 ] [8,14,26][8,14,26][8,14,26].
We now turn to Theorem 6.2, which asserts that torsion points generally do not lie on a subvariety. According to Table 1, for the dynamical analogue we should replace the torsion points with preperiodic points, leading to our second dynamical unlikely intersection conjecture.
Conjecture 6.8 (Dynamical Manin-Mumford conjecture). Let X / C X / C X//CX / \mathbb{C}X/C be a smooth quasiprojective variety, let f : X X f : X → X f:X rarr Xf: X \rightarrow Xf:X→X be a regular self-map of X X XXX, and let Y X Y ⊆ X Y sube XY \subseteq XY⊆X be a subvariety of X X XXX. Then
PrePer ( f ) Y PrePer ⁡ ( f ) ∩ Y PrePer(f)nn Y\operatorname{PrePer}(f) \cap YPrePer⁡(f)∩Y is not Zariski dense in Y Y YYY (except when it "obviously" is).
Unfortunately, the following natural rigorous formulation of Conjecture 6.8 turns out to be false.
Incorrect Rigorous Formulation of Conjecture 6.8.

See [30] for a counterexample, and for an alternative formulation of Conjecture 6.8 that requires more stringent hypotheses on f f fff and Y Y YYY.
Both the Mumford-Manin and Mordell-Lang conjectures concern how special points lie on subvarieties of a given variety. The André-Oort conjecture has a similar flavor,
but the ambient variety is a moduli space and the specialness of the points comes from the properties of the objects that they represent. The André-Oort conjecture is easy to state as long as we are willing to sweep some quite technical definitions under the rug! 26 26 ^(26){ }^{26}26
Conjecture 6.9 (André-Oort conjecture). Let S S SSS be a Shimura variety, let Γ S Γ ⊂ S Gamma subS\Gamma \subset \mathcal{S}Γ⊂S be a set of special points of S S SSS, and let Y S Y ⊂ S Y sub SY \subset SY⊂S be an irreducible subvariety such that Γ Y Γ ∩ Y Gamma nn Y\Gamma \cap YΓ∩Y is Zariski dense in Y Y YYY. Then Y Y YYY is a special subvariety of S S SSS.
The rough idea is that S S SSS is a moduli space whose points classify a certain class of abelian varieties, a collection of special points T S T ⊂ S Tsub S\mathcal{T} \subset ST⊂S consists of points whose associated abelian varieties have an additional special structure, and a special subvariety is one in which every associated abelian variety has the T T T\mathcal{T}T property for geometric reasons. The André-Oort conjecture has been proven in many cases, including for S = A 1 d S = A 1 d S=A_(1)^(d)S=\mathcal{A}_{1}^{d}S=A1d [70] and for S = A g S = A g S=A_(g)S=\mathcal{A}_{g}S=Ag [71,88].
We describe two sample dynamical unlikely intersection theorems that take place in the moduli space of unicritical polynomials, which are polynomials of the form x d + c x d + c x^(d)+cx^{d}+cxd+c. We view the first as a mixed unlikely intersection, because it involves one moduli parameter and two orbit parameters.
Theorem 6.10 ([3]). Let d 2 d ≥ 2 d >= 2d \geq 2d≥2, and let a , b C a , b ∈ C a,b inCa, b \in \mathbb{C}a,b∈C be complex numbers with a 2 b 2 a 2 ≠ b 2 a^(2)!=b^(2)a^{2} \neq b^{2}a2≠b2. Then
{ c C moduli parameter : a and b are both preperiodic special orbit parameters { c ∈ C ⏟ moduli parameter  : a  and  b  are both preperiodic  ⏟ special orbit parameters  {ubrace(c inCubrace)_("moduli parameter "):ubrace(a" and "b" are both preperiodic "ubrace)_("special orbit parameters ")\{\underbrace{c \in \mathbb{C}}_{\text {moduli parameter }}: \underbrace{a \text { and } b \text { are both preperiodic }}_{\text {special orbit parameters }}{c∈C⏟moduli parameter :a and b are both preperiodic ⏟special orbit parameters  for x d + c } x d + c } x^(d)+c}quadx^{d}+c\} \quadxd+c} is a finite set.
The second result has more of the flavor of the André-Oort conjecture in that it involves only moduli parameters and follows the dictionary in Table 1 by replacing complex multiplication abelian varieties with postcritically finite rational maps.
Theorem 6.11 ([28]). Let d 2 d ≥ 2 d >= 2d \geq 2d≥2, and let Y A 2 Y ⊂ A 2 Y subA^(2)Y \subset \mathbb{A}^{2}Y⊂A2 be an irreducible curve that is not a line of one of the following forms:
vertical line { ( a , t ) : t A 1 } ; ( a , t ) : t ∈ A 1 ; {(a,t):t inA^(1)};quad\left\{(a, t): t \in \mathbb{A}^{1}\right\} ; \quad{(a,t):t∈A1}; horizontal line { ( t , b ) : t A 1 } ; ( t , b ) : t ∈ A 1 ; {(t,b):t inA^(1)};\left\{(t, b): t \in \mathbb{A}^{1}\right\} ;{(t,b):t∈A1}; shifted diagonal line { ( t , ζ t ) : t A 1 } ( t , ζ t ) : t ∈ A 1 {(t,zeta t):t inA^(1)}\left\{(t, \zeta t): t \in \mathbb{A}^{1}\right\}{(t,ζt):t∈A1}, where ζ d 1 = 1 ζ d − 1 = 1 zeta^(d-1)=1\zeta^{d-1}=1ζd−1=1.
Then
{ ( a , b ) Y : x 2 + a and x 2 + b are both P C F special moduli parameters } is a finite set. { ( a , b ) ∈ Y : x 2 + a  and  x 2 + b  are both  P C F ⏟ special moduli parameters  }  is a finite set.  {ubrace((a,b)in Y:x^(2)+a" and "x^(2)+b" are both "PCFubrace)_("special moduli parameters ")}" is a finite set. "\{\underbrace{(a, b) \in Y: x^{2}+a \text { and } x^{2}+b \text { are both } P C F}_{\text {special moduli parameters }}\} \text { is a finite set. }{(a,b)∈Y:x2+a and x2+b are both PCF⏟special moduli parameters } is a finite set. 
A conjectural generalization of Theorem 6.10 allows both the map x 2 + c x 2 + c x^(2)+cx^{2}+cx2+c and the points a a aaa and b b bbb to vary simultaneously.
Conjecture 6.12 ([15,27]). Let d 2 d ≥ 2 d >= 2d \geq 2d≥2, let T T TTT be an irreducible curve, and let
α : T P 1 , β : T P 1 , and f : T End d 1 α : T → P 1 , β : T → P 1 ,  and  f : T → End d 1 alpha:T rarrP^(1),quad beta:T rarrP^(1),quad" and "quad f:T rarrEnd_(d)^(1)\alpha: T \rightarrow \mathbb{P}^{1}, \quad \beta: T \rightarrow \mathbb{P}^{1}, \quad \text { and } \quad f: T \rightarrow \operatorname{End}_{d}^{1}α:T→P1,β:T→P1, and f:T→Endd1
be morphisms, i.e., α α alpha\alphaα and β β beta\betaβ are 1-parameter families of points in P 1 P 1 P^(1)\mathbb{P}^{1}P1 and f f fff is a 1-parameter
26 See, for example, [89] for the precise definition of Shimura variety, special point, and special subvariety.
family of degree-d endomorphisms of P 1 P 1 P^(1)\mathbb{P}^{1}P1. Assume that the families α α alpha\alphaα and β β beta\betaβ are not f f fff dynamically related. 27 27 ^(27){ }^{27}27 Then
{ t T : α t and β t are both preperiodic for f t } is a finite set. t ∈ T : α t  and  β t  are both preperiodic for  f t  is a finite set.  {t in T:alpha_(t)" and "beta_(t)" are both preperiodic for "f_(t)}quad" is a finite set. "\left\{t \in T: \alpha_{t} \text { and } \beta_{t} \text { are both preperiodic for } f_{t}\right\} \quad \text { is a finite set. }{t∈T:αt and βt are both preperiodic for ft} is a finite set. 
Formulating a general dynamical André-Oort conjecture is more complicated. The first step is to construct an appropriate moduli space of rational maps with marked critical points: 28 28 ^(28){ }^{28}28
M d crit := { ( f , P 1 , , P 2 d 2 ) : f End d 1 and P 1 , , P 2 d 2 are critical points of f } / P G L 2 M d crit  := f , P 1 , … , P 2 d − 2 : f ∈ End d 1 ⁡  and  P 1 , … , P 2 d − 2  are critical points of  f / P G L 2 M_(d)^("crit "):={(f,P_(1),dots,P_(2d-2)):[f inEnd_(d)^(1)" and "P_(1)","dots","P_(2d-2)],[" are critical points of "f]}//PGL_(2)\mathcal{M}_{d}^{\text {crit }}:=\left\{\left(f, P_{1}, \ldots, P_{2 d-2}\right): \begin{array}{l} f \in \operatorname{End}_{d}^{1} \text { and } P_{1}, \ldots, P_{2 d-2} \\ \text { are critical points of } f \end{array}\right\} / \mathrm{PGL}_{2}Mdcrit :={(f,P1,…,P2d−2):f∈Endd1⁡ and P1,…,P2d−2 are critical points of f}/PGL2
Conjecture 6.13 (Dynamical André-Oort Conjecture, [4,82]). Let Y M d crit Y ⊆ M d crit  Y subeM_(d)^("crit ")Y \subseteq \mathcal{M}_{d}^{\text {crit }}Y⊆Mdcrit  be an algebraic subvariety such that the PCF maps in Y Y YYY are Zariski dense in Y Y YYY. Then Y Y YYY is cut out by "critical orbit relations."
Formulas of the form f n ( P i ) = f m ( P j ) f n P i = f m P j f^(n)(P_(i))=f^(m)(P_(j))f^{n}\left(P_{i}\right)=f^{m}\left(P_{j}\right)fn(Pi)=fm(Pj) define critical point relations, 29 29 ^(29){ }^{29}29 but other relations may arise from symmetries of f f fff, and even subtler relations may come from "hidden relations" due to subdynamical systems. See [82, REMARK 6.58] for an example due to Ingram. Thus for now we do not have a good geometric description of the phrase "critical orbit relations" in general, but there is such a description for 1-dimensional families, i.e., for Conjecture 6.13 with dim ( Y ) = 1 dim ⁡ ( Y ) = 1 dim(Y)=1\operatorname{dim}(Y)=1dim⁡(Y)=1 [4]. In this case the conjecture has been proven for 1dimensional families of polynomials [24], but it remains open for rational maps.

7. TOPIC #4: DYNATOMIC AND ARBOREAL REPRESENTATIONS

The focus of this section is on the arithmetic of fields generated by the coordinates of dynamically interesting points. We let K / Q K / Q K//QK / \mathbb{Q}K/Q be a number field, and we start with a motivating result from arithmetic geometry. Let E / K E / K E//KE / KE/K be an elliptic curve, and let
(7.1) ρ E / K , e l l : Gal ( K ¯ / K ) Aut ( T ( E ) ) G L 2 ( Z ) (7.1) ρ E / K , â„“ e l l : Gal ⁡ ( K ¯ / K ) → Aut ⁡ T â„“ ( E ) ≅ G L 2 Z â„“ {:(7.1)rho_(E//K,â„“)^(ell):Gal( bar(K)//K)rarr Aut(T_(â„“)(E))~=GL_(2)(Z_(â„“)):}\begin{equation*} \rho_{E / K, \ell}^{\mathrm{ell}}: \operatorname{Gal}(\bar{K} / K) \rightarrow \operatorname{Aut}\left(T_{\ell}(E)\right) \cong \mathrm{GL}_{2}\left(\mathbb{Z}_{\ell}\right) \tag{7.1} \end{equation*}(7.1)ρE/K,â„“ell:Gal⁡(K¯/K)→Aut⁡(Tâ„“(E))≅GL2(Zâ„“)
be the representation that describes the action of the Galois group on the â„“ â„“\ellâ„“-power torsion points of E E EEE. A famous theorem characterizes the image. 30 30 ^(30){ }^{30}30
28 It is easy to construct the GIT quotient for maps f f fff having 2 d 2 2 d − 2 2d-22 d-22d−2 distinct marked critical points, but some care is needed to handle maps having higher multiplicity critical points; see [20].
One might view these f n ( P i ) = f m ( P j ) f n P i = f m P j f^(n)(P_(i))=f^(m)(P_(j))f^{n}\left(P_{i}\right)=f^{m}\left(P_{j}\right)fn(Pi)=fm(Pj) relations as dynamical analogues of Hecke correspondences, although the analogy is somewhat tenuous.
Theorem 7.1 (Serre's Image-of-Galois Theorem, [77,78]). Assume that E does not have complex multiplication.
(a) For all sufficiently large primes â„“ â„“\ellâ„“, the Galois representation ρ E / K , e l l ρ E / K , â„“ e l l rho_(E//K,â„“)^(ell)\rho_{E / K, \ell}^{\mathrm{ell}}ρE/K,â„“ell is surjective.
(b) For all primes â„“ â„“\ellâ„“, the image of the Galois representation ρ E / K , e l l ρ E / K , â„“ e l l rho_(E//K,â„“)^(ell)\rho_{E / K, \ell}^{\mathrm{ell}}ρE/K,â„“ell is a subgroup of finite index in G L 2 ( Z ) G L 2 Z â„“ GL_(2)(Z_(â„“))\mathrm{GL}_{2}\left(\mathbb{Z}_{\ell}\right)GL2(Zâ„“).
There are analogous conjectures, and some theorems, for the Galois representations associated to higher-dimensional abelian varieties. We consider two analogues in arithmetic dynamics.

7.1. Topic #4(a): Dynatomic representations

Let
f : P N P N f : P N → P N f:P^(N)rarrP^(N)f: \mathbb{P}^{N} \rightarrow \mathbb{P}^{N}f:PN→PN
be a morphism of degree d 2 d ≥ 2 d >= 2d \geq 2d≥2 defined over K K KKK, and let
Per n ( f ) = { P P N ( K ¯ ) : P is f -periodic with exact period n } Per n ∗ ⁡ ( f ) = P ∈ P N ( K ¯ ) : P  is  f -periodic with exact period  n Per_(n)^(**)(f)={P inP^(N)(( bar(K))):P" is "f"-periodic with exact period "n}\operatorname{Per}_{n}^{*}(f)=\left\{P \in \mathbb{P}^{N}(\bar{K}): P \text { is } f \text {-periodic with exact period } n\right\}Pern∗⁡(f)={P∈PN(K¯):P is f-periodic with exact period n}
The action of f f fff on Per n ( f ) Per n ∗ ⁡ ( f ) Per_(n)^(**)(f)\operatorname{Per}_{n}^{*}(f)Pern∗⁡(f) splits it into a disjoint union of directed n n nnn-cycles, and the action of Gal ( K ¯ / K ) Gal ⁡ ( K ¯ / K ) Gal( bar(K)//K)\operatorname{Gal}(\bar{K} / K)Gal⁡(K¯/K) on Per n ( f ) Per n ∗ ⁡ ( f ) Per_(n)^(**)(f)\operatorname{Per}_{n}^{*}(f)Pern∗⁡(f) respects the cycle structure. The analogue of G L 2 G L 2 GL_(2)\mathrm{GL}_{2}GL2 in (7.1) is thus the group of automorphisms of the graph
P n , ν = a disjoint union of v directed n -gons. P n , ν =  a disjoint union of  v  directed  n -gons.  P_(n,nu)=" a disjoint union of "v" directed "n"-gons. "\mathscr{P}_{n, \nu}=\text { a disjoint union of } v \text { directed } n \text {-gons. }Pn,ν= a disjoint union of v directed n-gons. 
The abstract automorphism group of the directed graph P n , v P n , v P_(n,v)\mathscr{P}_{n, v}Pn,v is naturally described as a wreath product in which an automorphism of P n , v P n , v P_(n,v)\mathscr{P}_{n, v}Pn,v is characterized as a permutation of the v v vvv polygons combined with a rotation of each polygon:
Aut ( P n , v ) ( Z / n Z ) S v ( Z / n Z ) v S v Aut ⁡ P n , v ≅ ( Z / n Z ) S v ≅ ( Z / n Z ) v ⋊ S v Aut(P_(n,v))~=(Z//nZ)(:S_(v)~=(Z//nZ)^(v)><|S_(v):}\operatorname{Aut}\left(\mathcal{P}_{n, v}\right) \cong(\mathbb{Z} / n \mathbb{Z})\left\langle S_{v} \cong(\mathbb{Z} / n \mathbb{Z})^{v} \rtimes S_{v}\right.Aut⁡(Pn,v)≅(Z/nZ)⟨Sv≅(Z/nZ)v⋊Sv
Definition 7.2. Let f End N d ( K ) f ∈ End N d ⁡ ( K ) f inEnd_(N)^(d)(K)f \in \operatorname{End}_{N}^{d}(K)f∈EndNd⁡(K). The n n nnn-level dynatomic representation of f f fff over K K KKK is the homomorphism
ρ K , n , f d y n : Gal ( K ¯ / K ) Aut ( P n , ν ( f ) ) , where ν ( f ) = 1 n # Per n ( f ) ρ K , n , f d y n : Gal ⁡ ( K ¯ / K ) → Aut ⁡ P n , ν ( f ) ,  where  ν ( f ) = 1 n # Per n ∗ ⁡ ( f ) rho_(K,n,f)^(dyn):Gal( bar(K)//K)rarr Aut(P_(n,nu(f))),quad" where "nu(f)=(1)/(n)#Per_(n)^(**)(f)\rho_{K, n, f}^{\mathrm{dyn}}: \operatorname{Gal}(\bar{K} / K) \rightarrow \operatorname{Aut}\left(\mathcal{P}_{n, \nu(f)}\right), \quad \text { where } \nu(f)=\frac{1}{n} \# \operatorname{Per}_{n}^{*}(f)ρK,n,fdyn:Gal⁡(K¯/K)→Aut⁡(Pn,ν(f)), where ν(f)=1n#Pern∗⁡(f)
The analogue of Serre's theorem would assert that if f f fff has no automorphisms, 31 31 ^(31){ }^{31}31 then ρ K , n , f d y n ρ K , n , f d y n rho_(K,n,f)^(dyn)\rho_{K, n, f}^{\mathrm{dyn}}ρK,n,fdyn is surjective for sufficiently large n n nnn. It seems too much to ask that this be true for all maps, so we pose the following challenge:
Question 7.3 (Dynatomic Image-of-Galois Problem). Let K / Q K / Q K//QK / \mathbb{Q}K/Q be a number field, let N 1 N ≥ 1 N >= 1N \geq 1N≥1, and let d 2 d ≥ 2 d >= 2d \geq 2d≥2. Characterize the maps f End d N ( K ) f ∈ End d N ⁡ ( K ) f inEnd_(d)^(N)(K)f \in \operatorname{End}_{d}^{N}(K)f∈EnddN⁡(K) for which there is a constant C ( f ) C ( f ) C(f)C(f)C(f) such that for all n 1 n ≥ 1 n >= 1n \geq 1n≥1,
Image ( ρ K , f , n d y n ) has index at most C ( f ) in Aut ( P n , v ( f ) )  Image  ρ K , f , n d y n  has index at most  C ( f )  in  Aut ⁡ P n , v ( f ) " Image "(rho_(K,f,n)^(dyn))" has index at most "C(f)" in "Aut(P_(n,v(f)))\text { Image }\left(\rho_{K, f, n}^{\mathrm{dyn}}\right) \text { has index at most } C(f) \text { in } \operatorname{Aut}\left(\mathscr{P}_{n, v(f)}\right) Image (ρK,f,ndyn) has index at most C(f) in Aut⁡(Pn,v(f))
31 The automorphism group of f f fff is Aut ( f ) = { φ PGL N + 1 : φ 1 f φ = f } Aut ⁡ ( f ) = φ ∈ PGL N + 1 : φ − 1 ∘ f ∘ φ = f Aut(f)={varphi inPGL_(N+1):varphi^(-1)@f@varphi=f}\operatorname{Aut}(f)=\left\{\varphi \in \operatorname{PGL}_{N+1}: \varphi^{-1} \circ f \circ \varphi=f\right\}Aut⁡(f)={φ∈PGLN+1:φ−1∘f∘φ=f}. The elements of Gal ( K ¯ / K ) Gal ⁡ ( K ¯ / K ) Gal( bar(K)//K)\operatorname{Gal}(\bar{K} / K)Gal⁡(K¯/K) commute with the action of Aut K ( f ) Aut K ⁡ ( f ) Aut_(K)(f)\operatorname{Aut}_{K}(f)AutK⁡(f), so if Aut K ( f ) ( 1 ) Aut K ⁡ ( f ) ≠ ( 1 ) Aut_(K)(f)!=(1)\operatorname{Aut}_{K}(f) \neq(1)AutK⁡(f)≠(1), then the image of ρ K , n , f d y n ρ K , n , f d y n rho_(K,n,f)^(dyn)\rho_{K, n, f}^{\mathrm{dyn}}ρK,n,fdyn is restricted, just as the image of ρ E / K , ell ρ E / K , â„“ ell  rho_(E//K,â„“)^("ell ")\rho_{E / K, \ell}^{\text {ell }}ρE/K,â„“ell  is restricted if E E EEE has CM.

7.2. Topic #4(b): Arboreal representations

The dynatomic extensions described in Section 7.1 are generated by points with finite orbits. In this section we consider arboreal extensions, which are extensions generated by backward orbits.
Example 7.4. We illustrate with the map f ( x ) = x d f ( x ) = x d f(x)=x^(d)f(x)=x^{d}f(x)=xd.
(7.2) [ Dynatomic extension. Field generated by roots of x d n = x for n 1 Arboreal extension. Field generated by roots of x d n = a for n 1 ] (7.2)  Dynatomic extension. Field generated by roots of  x d n = x  for  n ≥ 1  Arboreal extension. Field generated by roots of  x d n = a  for  n ≥ 1 {:(7.2)[[" Dynatomic extension. Field generated by roots of "x^(d^(n))=x" for "n >= 1],[" Arboreal extension. Field generated by roots of "x^(d^(n))=a" for "n >= 1]]:}\left[\begin{array}{r} \text { Dynatomic extension. Field generated by roots of } x^{d^{n}}=x \text { for } n \geq 1 \tag{7.2}\\ \text { Arboreal extension. Field generated by roots of } x^{d^{n}}=a \text { for } n \geq 1 \end{array}\right](7.2)[ Dynatomic extension. Field generated by roots of xdn=x for n≥1 Arboreal extension. Field generated by roots of xdn=a for n≥1]
Thus (7.2) suggests that dynatomic extensions resemble cyclotomic extensions, while the arboreal extensions resemble Kummer extensions; although we readily admit that this is far from a perfect analogy.
Definition 7.5. Let f : P N P N f : P N → P N f:P^(N)rarrP^(N)f: \mathbb{P}^{N} \rightarrow \mathbb{P}^{N}f:PN→PN be a morphism of degree d 2 d ≥ 2 d >= 2d \geq 2d≥2 defined over K K KKK, and let P P N ( K ) P ∈ P N ( K ) P inP^(N)(K)P \in \mathbb{P}^{N}(K)P∈PN(K). The inverse image tree of f f fff rooted at P P PPP is the (disjoint) union of the inverse images of P P PPP by the iterates of f f fff :
T f , P = n 0 f n ( P ) = n 0 { Q P N ( K ¯ ) : f n ( Q ) = P } T f , P = ⋃ n ≥ 0   f − n ( P ) = ⋃ n ≥ 0   Q ∈ P N ( K ¯ ) : f n ( Q ) = P T_(f,P)=uuu_(n >= 0)f^(-n)(P)=uuu_(n >= 0){Q inP^(N)(( bar(K))):f^(n)(Q)=P}\mathcal{T}_{f, P}=\bigcup_{n \geq 0} f^{-n}(P)=\bigcup_{n \geq 0}\left\{Q \in \mathbb{P}^{N}(\bar{K}): f^{n}(Q)=P\right\}Tf,P=⋃n≥0f−n(P)=⋃n≥0{Q∈PN(K¯):fn(Q)=P}
We say that f f fff is arboreally complete at P P PPP if # f n ( P ) = d n N # f − n ( P ) = d n N #f^(-n)(P)=d^(nN)\# f^{-n}(P)=d^{n N}#f−n(P)=dnN for all n 0 n ≥ 0 n >= 0n \geq 0n≥0, in which case T f , P T f , P T_(f,P)\mathcal{T}_{f, P}Tf,P is a complete rooted d N d N d^(N)d^{N}dN-ary tree, where f f fff maps the points in f n 1 ( P ) f − n − 1 ( P ) f^(-n-1)(P)f^{-n-1}(P)f−n−1(P) to the points in f n ( P ) f − n ( P ) f^(-n)(P)f^{-n}(P)f−n(P). Figure 2 illustrates a complete inverse image tree for a degree-2 map f : P 1 P 1 f : P 1 → P 1 f:P^(1)rarrP^(1)f: \mathbb{P}^{1} \rightarrow \mathbb{P}^{1}f:P1→P1.

FIGURE 2

A complete binary inverse image tree
The points in the iterated inverse image of P P PPP generate a (generally infinite) algebraic extension of K K KKK, so the Galois group Gal ( K ¯ / K ) Gal ⁡ ( K ¯ / K ) Gal( bar(K)//K)\operatorname{Gal}(\bar{K} / K)Gal⁡(K¯/K) acts on the points in T f , P T f , P T_(f,P)\mathcal{T}_{f, P}Tf,P. And since the action of the Galois group commutes with the map f f fff, the action of Gal ( K ¯ / K ) Gal ⁡ ( K ¯ / K ) Gal( bar(K)//K)\operatorname{Gal}(\bar{K} / K)Gal⁡(K¯/K) on T f , P T f , P T_(f,P)\mathcal{T}_{f, P}Tf,P preserves the tree structure. Thus in this case, the analogue of G L 2 G L 2 GL_(2)\mathrm{GL}_{2}GL2 in (7.1) is the group of automorphisms of the tree T f , P T f , P T_(f,P)\mathcal{T}_{f, P}Tf,P, which leads us to our primary object of study.
Definition 7.6. Let f End N d ( K ) f ∈ End N d ⁡ ( K ) f inEnd_(N)^(d)(K)f \in \operatorname{End}_{N}^{d}(K)f∈EndNd⁡(K), and let P P N ( K ) P ∈ P N ( K ) P inP^(N)(K)P \in \mathbb{P}^{N}(K)P∈PN(K). The arboreal representation (over K K KKK ) of f f fff rooted at P P PPP is the homomorphism
ρ K , f , P d y n : Gal ( K ¯ / K ) Aut ( T f , P ) ρ K , f , P d y n : Gal ⁡ ( K ¯ / K ) → Aut ⁡ T f , P rho_(K,f,P)^(dyn):Gal( bar(K)//K)rarr Aut(T_(f,P))\rho_{K, f, P}^{\mathrm{dyn}}: \operatorname{Gal}(\bar{K} / K) \rightarrow \operatorname{Aut}\left(\mathcal{T}_{f, P}\right)ρK,f,Pdyn:Gal⁡(K¯/K)→Aut⁡(Tf,P)
The Odoni 32 32 ^(32){ }^{32}32 index over K K KKK of f f fff at P P PPP is the index of the image in the full tree automorphism group,
ι K ( f , P ) = [ Aut ( T f , P ) : Image ( ρ K , f , P d y n ) ] ι K ( f , P ) = Aut ⁡ T f , P : Image ⁡ ρ K , f , P d y n iota_(K)(f,P)=[Aut(T_(f,P)):Image(rho_(K,f,P)^(dyn))]\iota_{K}(f, P)=\left[\operatorname{Aut}\left(\mathcal{T}_{f, P}\right): \operatorname{Image}\left(\rho_{K, f, P}^{\mathrm{dyn}}\right)\right]ιK(f,P)=[Aut⁡(Tf,P):Image⁡(ρK,f,Pdyn)]
As in the dynatomic case, it is again too much to hope that the image of ρ K , f , P dyn ρ K , f , P dyn  rho_(K,f,P)^("dyn ")\rho_{K, f, P}^{\text {dyn }}ρK,f,Pdyn  has finite index in Aut ( T f , P ) Aut ⁡ T f , P Aut(T_(f,P))\operatorname{Aut}\left(\mathcal{T}_{f, P}\right)Aut⁡(Tf,P) for all f f fff, but we might expect this to be true for most f f fff. This leads to a number of fundamental questions.
Question 7.7 (Arboreal Image-of-Galois Problem).
(a) Let K / Q K / Q K//QK / \mathbb{Q}K/Q be a number field, and let N 1 N ≥ 1 N >= 1N \geq 1N≥1 and d 2 d ≥ 2 d >= 2d \geq 2d≥2. Characterize the maps f End d N ( K ) f ∈ End d N ⁡ ( K ) f inEnd_(d)^(N)(K)f \in \operatorname{End}_{d}^{N}(K)f∈EnddN⁡(K) and points P P N ( K ) P ∈ P N ( K ) P inP^(N)(K)P \in \mathbb{P}^{N}(K)P∈PN(K) whose Odoni index ι K ( f , P ) ι K ( f , P ) iota_(K)(f,P)\iota_{K}(f, P)ιK(f,P) is finite, especially when f f fff is arboreally complete at P P PPP.
(b) (Generalized Odoni conjecture) For all number fields K / Q K / Q K//QK / \mathbb{Q}K/Q and all N 1 N ≥ 1 N >= 1N \geq 1N≥1 and d 2 d ≥ 2 d >= 2d \geq 2d≥2, does there exist a point P P N ( K ) P ∈ P N ( K ) P inP^(N)(K)P \in \mathbb{P}^{N}(K)P∈PN(K) and a map f End d N ( K ) f ∈ End d N ⁡ ( K ) f inEnd_(d)^(N)(K)f \in \operatorname{End}_{d}^{N}(K)f∈EnddN⁡(K) that is arboreally complete at P P PPP such that ι K ( f , P ) = 1 ι K ( f , P ) = 1 iota_(K)(f,P)=1\iota_{K}(f, P)=1ιK(f,P)=1 ?
(c) Fix a number field K / Q K / Q K//QK / \mathbb{Q}K/Q and integers N 1 N ≥ 1 N >= 1N \geq 1N≥1 and d 2 d ≥ 2 d >= 2d \geq 2d≥2. Is it true that ι K ( f , P ) = 1 ι K ( f , P ) = 1 iota_(K)(f,P)=1\iota_{K}(f, P)=1ιK(f,P)=1 for "almost all" pairs ( f , P ) ( f , P ) (f,P)(f, P)(f,P) in End d N ( K ) × P N ( K ) End d N ⁡ ( K ) × P N ( K ) End_(d)^(N)(K)xxP^(N)(K)\operatorname{End}_{d}^{N}(K) \times \mathbb{P}^{N}(K)EnddN⁡(K)×PN(K) for some appropriate sense of density?
Remark 7.8. Odoni's original conjecture was both more restrictive and stronger than Question 7.7(b) in that he considered only N = 1 N = 1 N=1N=1N=1 and polynomial maps. Odoni asked if for all K / Q K / Q K//QK / \mathbb{Q}K/Q and all d 2 d ≥ 2 d >= 2d \geq 2d≥2, there exists a degree- d d ddd monic polynomial f ( x ) K [ x ] f ( x ) ∈ K [ x ] f(x)in K[x]f(x) \in K[x]f(x)∈K[x] and a point α K α ∈ K alpha in K\alpha \in Kα∈K such that T f , α T f , α T_(f,alpha)\mathcal{T}_{f, \alpha}Tf,α is a complete d d ddd-ary tree and such that ι K ( f , α ) = 1 ι K ( f , α ) = 1 iota_(K)(f,alpha)=1\iota_{K}(f, \alpha)=1ιK(f,α)=1. Odoni's conjecture was proven over Q Q Q\mathbb{Q}Q for prime values of d d ddd in [45], and then in full generality in [85]. We mention that Odoni originally conjectured that the statement should hold for all Hilbertian fields, but this was recently resolved in the negative [36].
Remark 7.9. We close with the well-known observation that the automorphism group of an n n nnn-level complete rooted regular tree (labeling the levels 0 , 1 , 2 , , n 0 , 1 , 2 , … , n 0,1,2,dots,n0,1,2, \ldots, n0,1,2,…,n ) is an n n nnn-fold wreath product of the symmetric group. Hence if f f fff is arboreally complete at P P PPP, then the automorphism group of T f , P T f , P T_(f,P)\mathcal{T}_{f, P}Tf,P is the inverse limit
32 Named in honor of R. W. K. Odoni, who appears to have been the first to seriously study such problems in a series of papers [65-67], in one of which he proves that ι Q ( x 2 x + 1 , 0 ) = 1 ι Q x 2 − x + 1 , 0 = 1 iota_(Q)(x^(2)-x+1,0)=1\iota_{\mathbb{Q}}\left(x^{2}-x+1,0\right)=1ιQ(x2−x+1,0)=1.
The profinite group G ( K ¯ / K ) G ( K ¯ / K ) G( bar(K)//K)G(\bar{K} / K)G(K¯/K) then acts continuously on the profinite group Aut ( T f , P ) Aut ⁡ T f , P Aut(T_(f,P))\operatorname{Aut}\left(\mathcal{T}_{f, P}\right)Aut⁡(Tf,P), just as in arithmetic geometry G ( K ¯ / K ) G ( K ¯ / K ) G( bar(K)//K)G(\bar{K} / K)G(K¯/K) acts continuously on the Tate module T ( A ) = lim A [ n ] T â„“ ( A ) = lim ⟵   A â„“ n T_(â„“)(A)=lim longleftarrowA[â„“^(n)]T_{\ell}(A)=\underset{\longleftarrow}{\lim } A\left[\ell^{n}\right]Tâ„“(A)=lim⟵A[â„“n] of an abelian variety A / K A / K A//KA / KA/K.

8. TOPIC #5: DYNAMICAL AND ARITHMETIC COMPLEXITY

We informally define the complexity of a mathematical object to be a rough estimate for how much space it takes to store the object:
h ( X ) = complexity of object X # of basic storage units (e.g., bits, scalars) required to describe X . h ( X ) =  complexity of object  X ≍ #  of basic storage units (e.g., bits, scalars) required to describe  X . {:[h(X)=" complexity of object "X],[≍#" of basic storage units (e.g., bits, scalars) required to describe "X.]:}\begin{aligned} h(\mathcal{X}) & =\text { complexity of object } \mathcal{X} \\ & \asymp \# \text { of basic storage units (e.g., bits, scalars) required to describe } \mathcal{X} . \end{aligned}h(X)= complexity of object X≍# of basic storage units (e.g., bits, scalars) required to describe X.
Example 8.1. The complexity of a nonzero integer c Z c ∈ Z c inZc \in \mathbb{Z}c∈Z is the number of bits needed to describe c c ccc, so roughly log | c | log ⁡ | c | log |c|\log |c|log⁡|c|.
Example 8.2. The complexity of a nonzero polynomial f ( x ) K [ x ] f ( x ) ∈ K [ x ] f(x)in K[x]f(x) \in K[x]f(x)∈K[x] is the number of coefficients needed to describe f f fff, so roughly deg ( f ) deg ⁡ ( f ) deg(f)\operatorname{deg}(f)deg⁡(f).
For a sequence of objects X = ( X n ) n 1 X = X n n ≥ 1 X=(X_(n))_(n >= 1)\boldsymbol{X}=\left(\mathcal{X}_{n}\right)_{n \geq 1}X=(Xn)n≥1 whose complexity is expected to grow exponentially, we define the sequential complexity of X X X\boldsymbol{X}X to be the limit 33 limit 33 limit^(33)\operatorname{limit}^{33}limit33
σ ( x ) = lim n h ( X n ) 1 / n σ ( x ) = lim n → ∞   h X n 1 / n sigma(x)=lim_(n rarr oo)h(X_(n))^(1//n)\sigma(\boldsymbol{x})=\lim _{n \rightarrow \infty} h\left(\mathcal{X}_{n}\right)^{1 / n}σ(x)=limn→∞h(Xn)1/n
Example 8.3. Let f : P N P N f : P N → P N f:P^(N)rarrP^(N)f: \mathbb{P}^{N} \rightarrow \mathbb{P}^{N}f:PN→PN be a degree- d d ddd dominant rational map, i.e., a map given by homogeneous degree- d d ddd polynomials [ f 0 , , f N ] f 0 , … , f N [f_(0),dots,f_(N)]\left[f_{0}, \ldots, f_{N}\right][f0,…,fN] in C [ x 0 , , x N ] C x 0 , … , x N C[x_(0),dots,x_(N)]\mathbb{C}\left[x_{0}, \ldots, x_{N}\right]C[x0,…,xN] having no common factors. Then h ( f ) = deg ( f ) = d h ( f ) = deg ⁡ ( f ) = d h(f)=deg(f)=dh(f)=\operatorname{deg}(f)=dh(f)=deg⁡(f)=d. The sequential complexity of the sequence of iterates f n f n f^(n)f^{n}fn is called the dynamical degree of f f fff and is denoted
(8.1) δ f = lim n ( deg f n ) 1 / n (8.1) δ f = lim n → ∞   deg ⁡ f n 1 / n {:(8.1)delta_(f)=lim_(n rarr oo)(deg f^(n))^(1//n):}\begin{equation*} \delta_{f}=\lim _{n \rightarrow \infty}\left(\operatorname{deg} f^{n}\right)^{1 / n} \tag{8.1} \end{equation*}(8.1)δf=limn→∞(deg⁡fn)1/n
Example 8.4. Let P = [ c 0 , , c N ] P N ( Q ) P = c 0 , … , c N ∈ P N ( Q ) P=[c_(0),dots,c_(N)]inP^(N)(Q)P=\left[c_{0}, \ldots, c_{N}\right] \in \mathbb{P}^{N}(\mathbb{Q})P=[c0,…,cN]∈PN(Q) be a point written with relatively prime integer coordinates. Then
(8.2) h ( P ) = log max | c i | (8.2) h ( P ) = log ⁡ max c i {:(8.2)h(P)=log max|c_(i)|:}\begin{equation*} h(P)=\log \max \left|c_{i}\right| \tag{8.2} \end{equation*}(8.2)h(P)=log⁡max|ci|
More generally, if K / Q K / Q K//QK / \mathbb{Q}K/Q is a number field, then there is a well-defined Weil height function 34 34 ^(34){ }^{34}34
(8.3) h : P N ( K ) [ 0 , ) (8.3) h : P N ( K ) → [ 0 , ∞ ) {:(8.3)h:P^(N)(K)rarr[0","oo):}\begin{equation*} h: \mathbb{P}^{N}(K) \rightarrow[0, \infty) \tag{8.3} \end{equation*}(8.3)h:PN(K)→[0,∞)
33 In cases where the limit is not known to exist, we may consider the upper and lower sequential complexities
σ ¯ ( X ) = lim sup n h ( X n ) 1 / n and σ _ ( X ) = lim inf n h ( X n ) 1 / n σ ¯ ( X ) = lim sup n → ∞   h X n 1 / n  and  σ _ ( X ) = lim inf n → ∞   h X n 1 / n bar(sigma)(X)=l i m   s u p_(n rarr oo)h(X_(n))^(1//n)quad" and "quadsigma _(X)=l i m   i n f_(n rarr oo)h(X_(n))^(1//n)\bar{\sigma}(\boldsymbol{X})=\limsup _{n \rightarrow \infty} h\left(\boldsymbol{X}_{n}\right)^{1 / n} \quad \text { and } \quad \underline{\sigma}(\boldsymbol{X})=\liminf _{n \rightarrow \infty} h\left(\mathcal{X}_{n}\right)^{1 / n}σ¯(X)=lim supn→∞h(Xn)1/n and σ_(X)=lim infn→∞h(Xn)1/n
34 The Weil height of a point P = [ a 0 , , a N ] P N ( K ) P = a 0 , … , a N ∈ P N ( K ) P=[a_(0),dots,a_(N)]inP^(N)(K)P=\left[a_{0}, \ldots, a_{N}\right] \in \mathbb{P}^{N}(K)P=[a0,…,aN]∈PN(K) may be defined as follows: Let

dings of K K KKK. Then
h ( P ) = 1 d log | N K / Q ( B ) | + 1 d i = 1 d log max 0 j N | σ i ( a j ) | h ( P ) = 1 d log ⁡ N K / Q ( B ) + 1 d ∑ i = 1 d   log ⁡ max 0 ≤ j ≤ N   σ i a j h(P)=(1)/(d)log|N_(K//Q)(B)|+(1)/(d)sum_(i=1)^(d)log max_(0 <= j <= N)|sigma_(i)(a_(j))|h(P)=\frac{1}{d} \log \left|\mathrm{N}_{K / \mathbb{Q}}(\mathfrak{B})\right|+\frac{1}{d} \sum_{i=1}^{d} \log \max _{0 \leq j \leq N}\left|\sigma_{i}\left(a_{j}\right)\right|h(P)=1dlog⁡|NK/Q(B)|+1d∑i=1dlog⁡max0≤j≤N|σi(aj)|
that generalizes (8.2). The height of a point P P N ( K ) P ∈ P N ( K ) P inP^(N)(K)P \in \mathbb{P}^{N}(K)P∈PN(K) measures the complexity of the coordinates of P P PPP.
Now let K / Q K / Q K//QK / \mathbb{Q}K/Q be a number field, let P P N ( K ) P ∈ P N ( K ) P inP^(N)(K)P \in \mathbb{P}^{N}(K)P∈PN(K), and let f : P N P N f : P N → P N f:P^(N)rarrP^(N)f: \mathbb{P}^{N} \rightarrow \mathbb{P}^{N}f:PN→PN be a dominant rational map defined over K K KKK. Then the sequential complexity of the orbit O f ( P ) O f ( P ) O_(f)(P)\mathcal{O}_{f}(P)Of(P) is called the arithmetic degree of the f f fff-orbit of P P PPP and is denoted
(8.4) α f ( P ) = lim n h ( f n ( P ) ) 1 / n (8.4) α f ( P ) = lim n → ∞   h f n ( P ) 1 / n {:(8.4)alpha_(f)(P)=lim_(n rarr oo)h(f^(n)(P))^(1//n):}\begin{equation*} \alpha_{f}(P)=\lim _{n \rightarrow \infty} h\left(f^{n}(P)\right)^{1 / n} \tag{8.4} \end{equation*}(8.4)αf(P)=limn→∞h(fn(P))1/n
The notation in Table 2 will be used throughout the remainder of this section. We will generalize the complexity measures from Examples 8.3 and 8.4 and describe a number of results and questions.
Definition 8.5. The (first) dynamical degree of a dominant rational map f : X X f : X → X f:X rarr Xf: X \rightarrow Xf:X→X is
(8.5) δ f = lim n ( deg X ( f n ) ) 1 / n (8.5) δ f = lim n → ∞   deg X ⁡ f n 1 / n {:(8.5)delta_(f)=lim_(n rarr oo)(deg_(X)(f^(n)))^(1//n):}\begin{equation*} \delta_{f}=\lim _{n \rightarrow \infty}\left(\operatorname{deg}_{X}\left(f^{n}\right)\right)^{1 / n} \tag{8.5} \end{equation*}(8.5)δf=limn→∞(degX⁡(fn))1/n
The limit (8.5) converges and is independent of the choice of the ample divisor H H HHH used to define deg X [ 16 ] deg X ⁡ [ 16 ] deg_(X)[16]\operatorname{deg}_{X}[16]degX⁡[16]. 35 35 ^(35){ }^{35}35 Dynamical degrees on P N P N P^(N)\mathbb{P}^{N}PN were first studied in the 1990 s [ 2 , 9 , 75 ] [ 2 , 9 , 75 ] [2,9,75][2,9,75][2,9,75]. A long-standing question concerning the algebraicity of the dynamical degree was recently answered in the negative.
Theorem 8.6 ([5,6]). For all N 2 N ≥ 2 N >= 2N \geq 2N≥2, there exist dominant rational maps f : P N P N f : P N → P N f:P^(N)rarrP^(N)f: \mathbb{P}^{N} \rightarrow \mathbb{P}^{N}f:PN→PN defined over Q Q Q\mathbb{Q}Q such that δ f δ f delta_(f)\delta_{f}δf is a transcendental number. For N 3 N ≥ 3 N >= 3N \geq 3N≥3, there exist such maps that are birational automorphisms of P N P N P^(N)\mathbb{P}^{N}PN.
K K KKK a number field with algebraic closure K ¯ K ¯ bar(K)\bar{K}K¯
X X XXX a smooth projective variety of dimension d d ddd defined over K K KKK
f f fff a dominant rational map f : X X f : X → X f:X rarr Xf: X \rightarrow Xf:X→X defined over K K KKK
X f X f X_(f)X_{f}Xf = { P X ( K ¯ ) : f = P ∈ X ( K ¯ ) : f ={P in X(( bar(K))):f:}=\left\{P \in X(\bar{K}): f\right.={P∈X(K¯):f is well-defined at f n ( P ) f n ( P ) f^(n)(P)f^{n}(P)fn(P) all n 0 } n ≥ 0 {:n >= 0}\left.n \geq 0\right\}n≥0}
deg X ( f ) deg X ⁡ ( f ) deg_(X)(f)\operatorname{deg}_{X}(f)degX⁡(f) = ( f H ) H d 1 = f ∗ H ⋅ H d − 1 =(f^(**)H)*H^(d-1)=\left(f^{*} H\right) \cdot H^{d-1}=(f∗H)⋅Hd−1, where H H HHH is an ample divisor on X X XXX, and this formula
is a d d ddd-fold intersection index
the height on X X XXX coming from a projective embedding ι : X P N ι : X ↪ P N iota:X↪P^(N)\iota: X \hookrightarrow \mathbb{P}^{N}ι:X↪PN, i.e.,
h X h X h_(X)h_{X}hX
h X = h ι h X = h ∘ ι h_(X)=h@iotah_{X}=h \circ \iotahX=h∘ι, where h h hhh is the Weil height (8.3) on P N P N P^(N)\mathbb{P}^{N}PN
h X + h X + h_(X)^(+)h_{X}^{+}hX+
h_(X)=h@iota, where h is the Weil height (8.3) on P^(N) h_(X)^(+)| $h_{X}=h \circ \iota$, where $h$ is the Weil height (8.3) on $\mathbb{P}^{N}$ | | :--- | | | | $h_{X}^{+}$ |
K a number field with algebraic closure bar(K) X a smooth projective variety of dimension d defined over K f a dominant rational map f:X rarr X defined over K X_(f) ={P in X(( bar(K))):f:} is well-defined at f^(n)(P) all {:n >= 0} deg_(X)(f) =(f^(**)H)*H^(d-1), where H is an ample divisor on X, and this formula is a d-fold intersection index the height on X coming from a projective embedding iota:X↪P^(N), i.e., h_(X) "h_(X)=h@iota, where h is the Weil height (8.3) on P^(N) h_(X)^(+)"| $K$ | a number field with algebraic closure $\bar{K}$ | | :--- | :--- | | $X$ | a smooth projective variety of dimension $d$ defined over $K$ | | $f$ | a dominant rational map $f: X \rightarrow X$ defined over $K$ | | $X_{f}$ | $=\left\{P \in X(\bar{K}): f\right.$ is well-defined at $f^{n}(P)$ all $\left.n \geq 0\right\}$ | | $\operatorname{deg}_{X}(f)$ | $=\left(f^{*} H\right) \cdot H^{d-1}$, where $H$ is an ample divisor on $X$, and this formula | | is a $d$-fold intersection index | | | the height on $X$ coming from a projective embedding $\iota: X \hookrightarrow \mathbb{P}^{N}$, i.e., | | | $h_{X}$ | $h_{X}=h \circ \iota$, where $h$ is the Weil height (8.3) on $\mathbb{P}^{N}$ <br> <br> $h_{X}^{+}$ |

TABLE 2

Notation for Section 8
35 The convergence of (8.5) when X = P N X = P N X=P^(N)X=\mathbb{P}^{N}X=PN is a fun exercise using deg ( f g ) deg ⁡ ( f ∘ g ) ≤ deg(f@g) <=\operatorname{deg}(f \circ g) \leqdeg⁡(f∘g)≤ ( deg f ) ( deg g ) ( deg ⁡ f ) ( deg ⁡ g ) (deg f)(deg g)(\operatorname{deg} f)(\operatorname{deg} g)(deg⁡f)(deg⁡g).
There is an arithmetic analogue of the dynamical degree that measures the average arithmetic complexity of the algebraic points in an orbit. But since rational maps may not be defined everywhere, the next definition must restrict attention to X f X f X_(f)X_{f}Xf, the points in X X XXX where the full forward orbit of f f fff is well defined. 36 36 ^(36){ }^{36}36
Definition 8.7. Let f : X X f : X → X f:X rarr Xf: X \rightarrow Xf:X→X be a dominant rational map defined over K K KKK, and let P P ∈ P inP \inP∈ X f ( K ¯ ) X f ( K ¯ ) X_(f)( bar(K))X_{f}(\bar{K})Xf(K¯). The arithmetic degree of the f f fff-orbit of P P PPP is
(8.6) α f ( P ) = lim n h X + ( f n ( P ) ) 1 / n (8.6) α f ( P ) = lim n → ∞   h X + f n ( P ) 1 / n {:(8.6)alpha_(f)(P)=lim_(n rarr oo)h_(X)^(+)(f^(n)(P))^(1//n):}\begin{equation*} \alpha_{f}(P)=\lim _{n \rightarrow \infty} h_{X}^{+}\left(f^{n}(P)\right)^{1 / n} \tag{8.6} \end{equation*}(8.6)αf(P)=limn→∞hX+(fn(P))1/n
Question 8.8. Does the limit (8.6) always exist?
In any case, we may consider the upper and lower arithmetic degrees α _ f ( P ) α _ f ( P ) alpha __(f)(P)\underline{\alpha}_{f}(P)α_f(P) and α ¯ f ( P ) α ¯ f ( P ) bar(alpha)_(f)(P)\bar{\alpha}_{f}(P)α¯f(P) defined using, respectively, the liminf and the limsup. It is not hard to show that these quantities are independent of the choice of the complexity function h X + h X + h_(X)^(+)h_{X}^{+}hX+. It is also easy to show that α ¯ f ( P ) α ¯ f ( P ) bar(alpha)_(f)(P)\bar{\alpha}_{f}(P)α¯f(P) is finite, but more difficult to show that there is a uniform geometric bound, as in the next result.
Theorem 8.9 ([50]). Let f : X X f : X → X f:X rarr Xf: X \rightarrow Xf:X→X be a dominant rational map defined over K K KKK, and let P X f ( K ¯ ) P ∈ X f ( K ¯ ) P inX_(f)( bar(K))P \in X_{f}(\bar{K})P∈Xf(K¯). Then
α ¯ f ( P ) δ f α ¯ f ( P ) ≤ δ f bar(alpha)_(f)(P) <= delta_(f)\bar{\alpha}_{f}(P) \leq \delta_{f}α¯f(P)≤δf
Moral of Theorem 8.9. The arithmetic complexity of an orbit is no worse than the dynamical complexity of the map.
Theorem 8.9 suggests a natural question. When do the arithmetic and dynamical complexities coincide?
Conjecture 8.10 ([39,40])). Let f : X X f : X → X f:X rarr Xf: X \rightarrow Xf:X→X be a dominant rational map defined over K K KKK, and let P X f ( K ¯ ) P ∈ X f ( K ¯ ) P inX_(f)( bar(K))P \in X_{f}(\bar{K})P∈Xf(K¯). Then
O f ( P ) is Zariski dense in X α ¯ f ( P ) = δ f O f ( P )  is Zariski dense in  X ⟹ α ¯ f ( P ) = δ f O_(f)(P)" is Zariski dense in "X Longrightarrow bar(alpha)_(f)(P)=delta_(f)\mathcal{O}_{f}(P) \text { is Zariski dense in } X \Longrightarrow \bar{\alpha}_{f}(P)=\delta_{f}Of(P) is Zariski dense in X⟹α¯f(P)=δf
Moral of Conjecture 8.10. An orbit with maximal geometric complexity
also has maximal arithmetic complexity.
Question 8.11. Does X ( K ¯ ) X ( K ¯ ) X( bar(K))X(\bar{K})X(K¯) always contain a point with Zariski dense f f fff-orbit? The answer is clearly no. For example, if there exists a dominant rational map φ : X P 1 φ : X → P 1 varphi:X rarrP^(1)\varphi: X \rightarrow \mathbb{P}^{1}φ:X→P1 satisfying φ f = φ φ ∘ f = φ varphi@f=varphi\varphi \circ f=\varphiφ∘f=φ, then each f f fff-orbit lies in a fiber of φ φ varphi\varphiφ. Xie asks whether this is the only obstruction. An affirmative answer for certain maps in dimension 2 is given in [35,93].
Example 8.12. It is easy to prove Conjecture 8.10 for morphisms f f fff of P N P N P^(N)\mathbb{P}^{N}PN, since in that case δ f = deg ( f ) δ f = deg ⁡ ( f ) delta_(f)=deg(f)\delta_{f}=\operatorname{deg}(f)δf=deg⁡(f), and the theory of canonical heights implies that
α f ( P ) = { deg ( f ) if # O f ( P ) = 1 if P is f -prepediodic α f ( P ) = deg ⁡ ( f )  if  # O f ( P ) = ∞ 1  if  P  is  f -prepediodic  alpha_(f)(P)={[deg(f)," if "#O_(f)(P)=oo],[1," if "P" is "f"-prepediodic "]:}\alpha_{f}(P)= \begin{cases}\operatorname{deg}(f) & \text { if } \# \mathcal{O}_{f}(P)=\infty \\ 1 & \text { if } P \text { is } f \text {-prepediodic }\end{cases}αf(P)={deg⁡(f) if #Of(P)=∞1 if P is f-prepediodic 
More generally, a similar argument works for endomorphisms of any smooth projective variety whose Néron-Severi group has rank 1 [38]. But the conjecture is still open for dominant rational maps of P N P N P^(N)\mathbb{P}^{N}PN, and for morphisms of more general varieties.
Example 8.13. The past decade has been significant progress on various cases of Conjecture 8.10, especially in the case of morphisms, using an assortment of tools ranging from linear-forms-in-logarithms to canonical heights for nef divisors to the minimal model program in algebraic geometry. In particular, Conjecture 8.10 has been proven for
  • group endomorphisms (homomorphisms composed with translations) of semiabelian varieties (extensions of abelian varieties by algebraic tori) [39, 52, 83, 84],
  • endomorphisms of (not necessarily smooth) projective surfaces [38,53,57],
  • extensions to P N P N P^(N)\mathbb{P}^{N}PN of regular affine automorphisms of A N A N A^(N)\mathbb{A}^{N}AN [38],
  • endomorphisms of hyperkähler varieties [43],
  • endomorphisms of degree greater than 1 of smooth projective threefolds of Kodaira dimension 0 [43],
  • endomorphisms of normal projective varieties such that Pic 0 Q = 0 Pic 0 ⊗ Q = 0 Pic^(0)oxQ=0\operatorname{Pic}^{0} \otimes \mathbb{Q}=0Pic0⊗Q=0 and with nef cone generated by finitely many semi-ample integral divisors [49], and
  • smooth projective threefolds having at least one int-amplified 37 37 ^(37){ }^{37}37 endomorphism, and surjective endomorphisms of smooth rationally connected projective varieties [51].
Remark 8.14. Various generalizations of Conjecture 8.10 have been proposed. We mention in particular the Small Arithmetic Non-Density Conjecture [51], which says that points of small arithmetic degree are not Zariski dense when f f fff is a morphism. However, as the authors observe, their conjecture is only for morphisms, since it may fail for dominant rational maps. The authors of [51] prove the SAND conjecture for many of the cases listed in Example 8.13.
Conjecture 8.10 is a relatively coarse estimate for the height growth of points in Zariski-dense orbits. An affirmative answer to the following question would yield a quantitative version of the conjecture.
37 A morphism f : X X f : X → X f:X rarr Xf: X \rightarrow Xf:X→X is int-amplified if there exists an ample Cartier divisor H H HHH such that f H H f ∗ H − H f^(**)H-Hf^{*} H-Hf∗H−H is also ample.
Question 8.15 ([10, QUestion 14.5]). Let f : X X f : X → X f:X rarr Xf: X \rightarrow Xf:X→X be a dominant rational map defined over K K KKK, and let P X f ( K ¯ ) P ∈ X f ( K ¯ ) P inX_(f)( bar(K))P \in X_{f}(\bar{K})P∈Xf(K¯) be a point whose orbit O f ( P ) O f ( P ) O_(f)(P)\mathcal{O}_{f}(P)Of(P) is Zariski dense in X X XXX. Do there exist (integers) 0 f N 0 ≤ â„“ f ≤ N 0 <= â„“_(f) <= N0 \leq \ell_{f} \leq N0≤ℓf≤N and k f 0 k f ≥ 0 k_(f) >= 0k_{f} \geq 0kf≥0 such that
h ( f n ( P ) ) δ f n n f ( log n ) k f h f n ( P ) ≍ δ f n â‹… n â„“ f â‹… ( log ⁡ n ) k f h(f^(n)(P))≍delta_(f)^(n)*n^(â„“_(f))*(log n)^(k_(f))h\left(f^{n}(P)\right) \asymp \delta_{f}^{n} \cdot n^{\ell_{f}} \cdot(\log n)^{k_{f}}h(fn(P))≍δfnâ‹…nâ„“fâ‹…(log⁡n)kf
where the implied constants depend on f f fff and P P PPP, but not on n n nnn ? If δ f > 1 δ f > 1 delta_(f) > 1\delta_{f}>1δf>1, is it further true that k f = 0 k f = 0 k_(f)=0k_{f}=0kf=0 ?

ACKNOWLEDGMENTS

The author would like to thank Rob Benedetto, John Doyle, Nicole Looper, Michelle Manes, and Yohsuke Matsuzawa for their helpful comments, corrections, and suggestions.

FUNDING

This work was partially supported by Simons Collaboration Grant #712332.

REFERENCES

[1] E. Amerik, Existence of non-preperiodic algebraic points for a rational self-map of infinite order. Math. Res. Lett. 18 (2011), no. 2, 251-256.
[2] V. I. Arnol'd, Dynamics of complexity of intersections. Bol. Soc. Bras. Mat. (N. S.) 21 (1990), no. 1, 1-10.
[3] M. Baker and L. DeMarco, Preperiodic points and unlikely intersections. Duke Math. J. 159 (2011), no. 1, 1-29.
[4] M. Baker and L. De Marco, Special curves and postcritically finite polynomials. Forum Math. Pi 1 (2013), e3, 35 pp.
[5] J. Bell, J. Diller, M. Jonsson, and H. Krieger, Birational mpas with transcendental dynamical degree. 2021, arXiv:2107.04113.
[6] J. P. Bell, J. Diller, and M. Jonsson, A transcendental dynamical degree. Acta Math. 225 (2020), no. 2, 193-225.
[7] J. P. Bell, D. Ghioca, and T. J. Tucker, The dynamical Mordell-Lang problem for étale maps. Amer. J. Math. 132 (2010), no. 6, 1655-1675.
[8] J. P. Bell, D. Ghioca, and T. J. Tucker, The dynamical Mordell-Lang conjecture. Math. Surveys Monogr. 210, American Mathematical Society, Providence, RI, 2016.
[9] M. P. Bellon and C.-M. Viallet, Algebraic entropy. Comm. Math. Phys. 204 (1999), no. 2, 425-437.
[10] R. Benedetto, P. Ingram, R. Jones, M. Manes, J. H. Silverman, and T. J. Tucker, Current trends and open problems in arithmetic dynamics. Bull. Amer. Math. Soc. (N.S.) 56 (2019), no. 4, 611-685.
[11] R. L. Benedetto, D. Ghioca, P. Kurlberg, and T. J. Tucker, A case of the dynamical Mordell-Lang conjecture. Math. Ann. 352 (2012), no. 1, 1-26.
[12] J. Blanc, J. K. Canci, and N. D. Elkies, Moduli spaces of quadratic rational maps with a marked periodic point of small order. Int. Math. Res. Not. IMRN 23 (2015), 12459 12489 12459 − 12489 12459-1248912459-1248912459−12489.
[13] T. Bousch, Sur quelques problèmes de dynamique holomorphe. Ph.D. thesis, Université de Paris-Sud, Centre d'Orsay, 1992.
[14] P. Corvaja, D. Ghioca, T. Scanlon, and U. Zannier, The dynamical Mordell-Lang conjecture for endomorphisms of semiabelian varieties defined over fields of positive characteristic. J. Inst. Math. Jussieu 20 (2021), no. 2, 669-698.
[15] L. DeMarco, Bifurcations, intersections, and heights. Algebra Number Theory 10 (2016), no. 5, 1031-1056.
[16] T.-C. Dinh and N. Sibony, Une borne supérieure pour l'entropie topologique d'une application rationnelle. Ann. of Math. (2) 161 (2005), no. 3, 1637-1644.
[17] J. R. Doyle, H. Krieger, A. Obus, R. Pries, S. Rubinstein-Salzedo, and L. West, Reduction of dynatomic curves. Ergodic Theory Dynam. Systems 39 (2019), no. 10, 2717-2768.
[18] J. R. Doyle and B. Poonen, Gonality of dynatomic curves and strong uniform boundedness of preperiodic points. Compos. Math. 156 (2020), no. 4, 733-743.
[19] J. R. Doyle and J. H. Silverman, A uniform field-of-definition/field-of-moduli bound for dynamical systems on P N P N P^(N)\mathbb{P}^{N}PN. J. Number Theory 195 (2019), 1-22.
[20] J. R. Doyle and J. H. Silverman, Moduli spaces for dynamical systems with portraits. Illinois J. Math. 64 (2020), no. 3, 375-465.
[21] N. Fakhruddin, Boundedness results for periodic points on algebraic varieties. Proc. Indian Acad. Sci. Math. Sci. 111 (2001), no. 2, 173-178.
[22] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73 (1983), no. 3, 349-366.
[23] G. Faltings, Diophantine approximation on abelian varieties. Ann. of Math. (2) 133 (1991), 349-366.
[24] C. Favre and T. Gauthier, The arithmetic of polynomial dynamical pairs. 2020, arXiv:2004.13801.
[25] E. V. Flynn, B. Poonen, and E. F. Schaefer, Cycles of quadratic polynomials and rational points on a genus-2 curve. Duke Math. J. 90 (1997), no. 3, 435-463.
[26] D. Ghioca, The dynamical Mordell-Lang conjecture in positive characteristic. Trans. Amer. Math. Soc. 371 (2019), no. 2, 1151-1167.
[27] D. Ghioca, L.-C. Hsia, and T. J. Tucker, Preperiodic points for families of polynomials. Algebra Number Theory 7 (2013), no. 3, 701-732.
[28] D. Ghioca, H. Krieger, K. D. Nguyen, and H. Ye, The dynamical André-Oort conjecture: unicritical polynomials. Duke Math. J. 166 (2017), no. 1, 1-25.
[29] D. Ghioca and T. J. Tucker, Periodic points, linearizing maps, and the dynamical Mordell-Lang problem. J. Number Theory 129 (2009), no. 6, 1392-1403.
[30] D. Ghioca, T. J. Tucker, and S. Zhang, Towards a dynamical Manin-Mumford conjecture. Int. Math. Res. Not. IMRN 22 (2011), 5109-5122.
[31] D. Ghioca and J. Xie, The dynamical Mordell-Lang conjecture for skew-linear self-maps. Appendix by Michael Wibmer. Int. Math. Res. Not. IMRN 21 (2020), 7433-7453.
[32] R. A. Hidalgo, A simple remark on the field of moduli of rational maps. Q. J. Math. 65 (2014), no. 2, 627-635.
[33] B. Hutz, Effectivity of dynatomic cycles for morphisms of projective varieties using deformation theory. Proc. Amer. Math. Soc. 140 (2012), no. 10, 3507 3514 3507 − 3514 3507-35143507-35143507−3514.
[34] P. Ingram, R. Ramadas, and J. H. Silverman, Post-critically finite maps in P n P n P^(n)\mathbb{P}^{n}Pn for n 2 n ≥ 2 n >= 2n \geq 2n≥2 are sparse. 2019, arXiv:1910.11290.
[35] J. Jia, J. Xie, and D.-Q. Zhang, Surjective endomorphisms of projective surfaces the existence of infinitely many dense orbits. 2020, arXiv:2005.03628.
[36] B. Kadets, Large arboreal Galois representations. J. Number Theory 210 (2020), 416-430.
[37] S. Kamienny, Torsion points on elliptic curves and q q qqq-coefficients of modular forms. Invent. Math. 109 (1992), no. 2, 221-229.
[38] S. Kawaguchi and J. H. Silverman, Examples of dynamical degree equals arithmetic degree. Michigan Math. J. 63 (2014), no. 1, 41-63.
[39] S. Kawaguchi and J. H. Silverman, Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties. Trans. Amer. Math. Soc. 368 (2016), no. 7, 5009-5035.
[40] S. Kawaguchi and J. H. Silverman, On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties. J. Reine Angew. Math. 713 (2016), 21-48.
[41] E. Lau and D. Schleicher, Internal addresses in the Mandelbrot set and irreducibility of polynomials. Tech. Rep. 1994/19, 1994.
[42] C. G. Lee and J. H. Silverman, GIT stability of Hénon maps. Proc. Amer. Math. Soc. 148 (2020), no. 10, 4263-4272.
[43] J. Lesieutre and M. Satriano, Canonical heights on hyper-Kähler varieties and the Kawaguchi-Silverman conjecture. Int. Math. Res. Not. IMRN 10 (2021), 7677 7714 7677 − 7714 7677-77147677-77147677−7714.
[44] A. Levy, The space of morphisms on projective space. Acta Arith. 146 (2011), no. 1, 13-31.
[45] N. Looper, Dynamical Galois groups of trinomials and Odoni's conjecture. Bull. Lond. Math. Soc. 51 (2019), no. 2, 278-292.
[46] N. Looper, The uniform boundedness and dynamical Lang conjectures for polynomials. 2021, arXiv:2105.05240.
[47] N. R. Looper, Dynamical uniform boundedness and the a b c a b c abca b cabc-conjecture. Invent. Math. 225 (2021), no. 1, 1-44.
[48] M. Manes, Moduli spaces for families of rational maps on P 1 P 1 P^(1)\mathbb{P}^{1}P1. J. Number Theory 129 (2009), no. 7, 1623-1663.
[49] Y. Matsuzawa, Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties. Adv. Math. 366 (2020), 107086
[50] Y. Matsuzawa, On upper bounds of arithmetic degrees. Amer. J. Math. 142 (2020), no. 6, 1797-1820.
[51] Y. Matsuzawa, S. Meng, T. Shibata, and D.-Q. Zhang, Non-density of points of small arithmetic degrees. 2020, arXiv:2002.10976.
[52] Y. Matsuzawa and K. Sano, Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties. Ergodic Theory Dynam. Systems 40 (2020), no. 6, 1655 1672 1655 − 1672 1655-16721655-16721655−1672.
[53] Y. Matsuzawa, K. Sano, and T. Shibata, Arithmetic degrees and dynamical degrees of endomorphisms on surfaces. Algebra Number Theory 12 (2018), no. 7, 1635-1657.
[54] B. Mazur, Modular curves and the Eisenstein ideal. Publ. Math. Inst. Hautes Études Sci. 47 (1977), 33-186.
[55] C. T. McMullen, Families of rational maps and iterative root-finding algorithms. Ann. of Math. (2) 125 (1987), no. 3, 467-493.
[56] M. McQuillan, Division points on semi-abelian varieties. Invent. Math. 120 (1995), no. 1, 143-159.
[57] S. Meng and D.-Q. Zhang, Kawaguchi-Silverman conjecture for surjective endomorphisms. 2019, arXiv:1908.01605.
[58] L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124 (1996), no. 1-3, 437-449.
[59] J. Milnor, Geometry and dynamics of quadratic rational maps. Exp. Math. 2 (1993), no. 1, 37-83.
[60] P. Morton, Arithmetic properties of periodic points of quadratic maps. Acta Arith. 62 (1992), no. 4, 343-372.
[61] P. Morton, On certain algebraic curves related to polynomial maps. Compos. Math. 103 (1996), no. 3, 319-350.
[62] P. Morton and J. H. Silverman, Rational periodic points of rational functions. Int. Math. Res. Not. 2 (1994), 97-110.
[63] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, third edn. Ergeb. Math. Grenzgeb. (2) [Results Math. Relat. Areas (2)] 34, Springer, Berlin, 1994.
[64] W. Narkiewicz, On a class of monic binomials. Proc. Steklov Inst. Math. 280 (2013), no. suppl. 2, S65-S70.
[65] R. W. K. Odoni, The Galois theory of iterates and composites of polynomials. Proc. Lond. Math. Soc. (3) 51 (1985), no. 3, 385-414.
[66] R. W. K. Odoni, On the prime divisors of the sequence w n + 1 = 1 + w 1 w n w n + 1 = 1 + w 1 ⋯ w n w_(n+1)=1+w_(1)cdotsw_(n)w_{n+1}=1+w_{1} \cdots w_{n}wn+1=1+w1⋯wn. J. Lond. Math. Soc. (2) 32 (1985), no. 1, 1-11.
[67] R. W. K. Odoni, Realising wreath products of cyclic groups as Galois groups. Mathematika 35 (1988), no. 1, 101-113.
[68] C. Panraksa, Rational periodic points of x d + c x d + c x^(d)+cx^{d}+cxd+c and abc conjecture. 2021, arXiv:2105.03715.
[69] C. Petsche, L. Szpiro, and M. Tepper, Isotriviality is equivalent to potential good reduction for endomorphisms of P N P N P^(N)\mathbb{P}^{N}PN over function fields. J. Algebra 322 (2009), no. 9 , 3345 3365 9 , 3345 − 3365 9,3345-33659,3345-33659,3345−3365.
[70] J. Pila, o-minimality and the André-Oort conjecture for C n C n C^(n)\mathbb{C}^{n}Cn. Ann. of Math. (2) 173 (2011), 1779-1840.
[71] J. Pila and J. Tsimerman, Ax-Lindemann for A g A g A_(g)\mathcal{A}_{\mathrm{g}}Ag. Ann. of Math. (2) 179 (2014), no. 2, 659-681.
[72] B. Poonen, Uniform boundedness of rational points and preperiodic points. 2012, arXiv:1206.7104.
[73] M. Raynaud, Courbes sur une variété abélienne et points de torsion. Invent. Math. 71 (1983), no. 1, 207-233.
[74] M. Raynaud, Sous-variétés d'une variété abélienne et points de torsion. In Arithmetic and geometry, I, pp. 327-352, Progr. Math. 35, Birkhäuser Boston, Boston, MA, 1983.
[75] A. Russakovskii and B. Shiffman, Value distribution for sequences of rational mappings and complex dynamics. Indiana Univ. Math. J. 46 (1997), no. 3, 897-932.
[76] D. Schleicher, Internal addresses of the Mandelbrot set and Galois groups of polynomials. Arnold Math. J. 3 (2017), no. 1, 1-35.
[77] J.-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Invent. Math. 15 (1972), no. 4, 259-331.
[78] J.-P. Serre, Abelian l-adic representations and elliptic curves. Res. Notes Math. 7, A K Peters Ltd., Wellesley, MA, 1998.
[79] J. H. Silverman, Rational points on K3 surfaces: a new canonical height. Invent. Math. 105 (1991), no. 2, 347-373.
[80] J. H. Silverman, The space of rational maps on P 1 P 1 P^(1)\mathbb{P}^{1}P1. Duke Math. J. 94 (1998), no. 1, 41-77.
[81] J. H. Silverman, Height estimates for equidimensional dominant rational maps. J. Ramanujan Math. Soc. 26 (2011), no. 2, 145-163.
[82] J. H. Silverman, Moduli Spaces and Arithmetic Dynamics. CRM Monogr. Ser. 30, American Mathematical Society, Providence, RI, 2012.
[83] J. H. Silverman, Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space. Ergodic Theory Dynam. Systems 34 (2014), no. 2, 647-678.
[84] J. H. Silverman, Arithmetic and dynamical degrees on abelian varieties. J. Théor. Nombres Bordeaux 29 (2017), no. 1, 151-167.
[85] J. Specter, Polynomials with surjective arboreal Galois representations exist in every degree. 2018, arXiv:1803.00434.
[86] M. Stoll, Rational 6-cycles under iteration of quadratic polynomials. LMS J. Comput. Math. 11 (2008), 367-380.
[87] Y.-S. Tai, On the Kodaira dimension of the moduli space of abelian varieties. Invent. Math. 68 (1982), no. 3, 425-439.
[88] J. Tsimerman, The André-Oort conjecture for A g A g A_(g)\mathscr{A}_{g}Ag. Ann. of Math. (2) 187 (2018), no. 2, 379-390.
[89] E. Ullmo and A. Yafaev, Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture. Ann. of Math. (2) 180 (2014), no. 3, 823-865.
[90] P. Vojta, Siegel's theorem in the compact case. Ann. of Math. (2) 133 (1991), no. 3, 509-548.
[91] R. Walde and P. Russo, Rational periodic points of the quadratic function Q c ( x ) = x 2 + c Q c ( x ) = x 2 + c Q_(c)(x)=x^(2)+cQ_{c}(x)=x^{2}+cQc(x)=x2+c. Amer. Math. Monthly 101 (1994), no. 4, 318-331.
[92] J. Xie, The dynamical Mordell-Lang conjecture for polynomial endomorphisms of the affine plane. Astérisque (2017), no. 394, vi+110 pp.
[93] J. Xie, The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane. Compos. Math. 153 (2017), no. 8, 1658-1672.
[94] J. Xie, Remarks on algebraic dynamics in positive characteristic. 2021, arXiv:2107.03559.

JOSEPH H. SILVERMAN

Mathematics Department, Brown University, Box 1917, Providence, RI 02912, USA, joseph_silverman@brown.edu
  1. NUMBER THEORY

RELATIVE TRACE FORMULAE AND THE GAN-GROSS-PRASAD CONJECTURES

RAPHAËL BEUZART-PLESSIS

Abstract

This paper reports on some recent progress that have been made on the so-called GanGross-Prasad conjectures through the use of relative trace formulae. In their global aspects, these conjectures, as well as certain refinements first proposed by Ichino-Ikeda, give precise relations between the central values of some higher-rank L L LLL-functions and automorphic periods. There are also local counterparts describing branching laws between representations of classical groups. In both cases, approaches through relative trace formulae have shown to be very successful and have even lead to complete proofs, at least in the case of unitary groups. However, the works leading to these definite results have also been the occasion to develop further and gain new insights on these fundamental tools of the still emerging relative Langlands program.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11F70; Secondary 22E50, 22E55, 11F72

KEYWORDS

Gan-Gross-Prasad conjecture, relative trace formula, periods of automorphic forms, L-functions, fundamental lemma, endoscopy
In broad terms, the Gan-Gross-Prasad conjectures concern two interrelated questions in the fields of representation theory and automorphic forms. On the one hand, these conjectures predict highly-sophisticated descriptions of some branching laws between representations of classical groups (that is, orthogonal, symplectic/metaplectic, or unitary groups) over local fields which can be seen as direct descendants of classical results of H H H\mathrm{H}H. Weyl on similar branching problems for compact Lie groups. The predictions are given in terms of the recently established local Langlands correspondence for these groups that provides a parameterization of the irreducible representations in terms of data of arithmetic nature. On the other hand, the Gan-Gross-Prasad conjectures also give far-reaching higher-rank generalizations of certain celebrated relations between special values of L L LLL-functions and period integrals. We start this paper by discussing two, by now well-known, examples of the former kind of relations.
First, we briefly review Hecke's integral representation for L L LLL-functions of modular forms. Let S 2 ( Γ 1 ( N ) ) S 2 Γ 1 ( N ) S_(2)(Gamma_(1)(N))S_{2}\left(\Gamma_{1}(N)\right)S2(Γ1(N)) be the space of cuspidal modular form of weight 2 for the group
Γ 1 ( N ) := { γ S L 2 ( Z ) | γ ( 1 0 1 ) mod N } Γ 1 ( N ) := γ ∈ S L 2 ( Z ) γ ≡ 1 ⋆ 0 1 mod N Gamma_(1)(N):={gamma inSL_(2)(Z)|gamma-=([1,***],[0,1])mod N}\Gamma_{1}(N):=\left\{\gamma \in \mathrm{SL}_{2}(\mathbb{Z}) \left\lvert\, \gamma \equiv\left(\begin{array}{ll} 1 & \star \\ 0 & 1 \end{array}\right) \bmod N\right.\right\}Γ1(N):={γ∈SL2(Z)|γ≡(1⋆01)modN}
It consists in the holomorphic functions f : H C f : H → C f:HrarrCf: \mathbb{H} \rightarrow \mathbb{C}f:H→C, where H = { x + i y x , y R , y > 0 } H = { x + i y ∣ x , y ∈ R , y > 0 } H={x+iy∣x,y inR,y > 0}\mathbb{H}=\{x+i y \mid x, y \in \mathbb{R}, y>0\}H={x+iy∣x,y∈R,y>0} is Poincaré upper half-plane, satisfying the functional equation
(0.1) f ( a z + b c z + d ) = ( c z + d ) 2 f ( z ) , ( a b c d ) Γ 1 ( N ) (0.1) f a z + b c z + d = ( c z + d ) 2 f ( z ) , ∀ a b c d ∈ Γ 1 ( N ) {:(0.1)f((az+b)/(cz+d))=(cz+d)^(2)f(z)","quad AA([a,b],[c,d])inGamma_(1)(N):}f\left(\frac{a z+b}{c z+d}\right)=(c z+d)^{2} f(z), \quad \forall\left(\begin{array}{ll} a & b \tag{0.1}\\ c & d \end{array}\right) \in \Gamma_{1}(N)(0.1)f(az+bcz+d)=(cz+d)2f(z),∀(abcd)∈Γ1(N)
and that are "vanishing at the cusps," a condition imposing in some sense means that f f fff is rapidly decreasing modulo the above symmetries. Another more geometric way to describe S 2 ( Γ 1 ( N ) ) S 2 Γ 1 ( N ) S_(2)(Gamma_(1)(N))S_{2}\left(\Gamma_{1}(N)\right)S2(Γ1(N)) is as a space of holomorphic differential forms: for f S 2 ( Γ 1 ( N ) ) f ∈ S 2 Γ 1 ( N ) f inS_(2)(Gamma_(1)(N))f \in S_{2}\left(\Gamma_{1}(N)\right)f∈S2(Γ1(N)), the form ω f = f ( z ) d z ω f = f ( z ) d z omega_(f)=f(z)dz\omega_{f}=f(z) d zωf=f(z)dz descends to the open modular curve Y 1 ( N ) = Γ 1 ( N ) H Y 1 ( N ) = Γ 1 ( N ) ∖ H Y_(1)(N)=Gamma_(1)(N)\\HY_{1}(N)=\Gamma_{1}(N) \backslash \mathbb{H}Y1(N)=Γ1(N)∖H (a Riemann surface as soon as N > 3 N > 3 N > 3N>3N>3 ) and the vanishing at the cusps condition translates to the fact that this form extends holomorphically to the canonical compactification X 1 ( N ) X 1 ( N ) X_(1)(N)X_{1}(N)X1(N) of Y 1 ( N ) Y 1 ( N ) Y_(1)(N)Y_{1}(N)Y1(N). Moreover, the map f ω f f ↦ ω f f|->omega_(f)f \mapsto \omega_{f}f↦ωf yields an isomorphism S 2 ( Γ 1 ( N ) ) Ω 1 ( X 0 ( N ) ) S 2 Γ 1 ( N ) ≃ Ω 1 X 0 ( N ) S_(2)(Gamma_(1)(N))≃Omega^(1)(X_(0)(N))S_{2}\left(\Gamma_{1}(N)\right) \simeq \Omega^{1}\left(X_{0}(N)\right)S2(Γ1(N))≃Ω1(X0(N)).
It follows from the functional equation (0.1) that every f S 2 ( Γ 1 ( N ) ) f ∈ S 2 Γ 1 ( N ) f inS_(2)(Gamma_(1)(N))f \in S_{2}\left(\Gamma_{1}(N)\right)f∈S2(Γ1(N)) is periodic of period 1 and thus admits a Fourier expansion
(0.2) f = n 1 a n q n , q = e 2 i π z (0.2) f = ∑ n ⩾ 1   a n q n , q = e 2 i Ï€ z {:(0.2)f=sum_(n >= 1)a_(n)q^(n)","quad q=e^(2i pi z):}\begin{equation*} f=\sum_{n \geqslant 1} a_{n} q^{n}, \quad q=e^{2 i \pi z} \tag{0.2} \end{equation*}(0.2)f=∑n⩾1anqn,q=e2iÏ€z
where the fact that the sum is restricted to positive integers is part of the assumption that f f fff vanishes at the cusps. The Hecke L L LLL-function of f f fff is then defined as the Dirichlet series
L ( s , f ) = n 1 a n n s L ( s , f ) = ∑ n ⩾ 1   a n n s L(s,f)=sum_(n >= 1)(a_(n))/(n^(s))L(s, f)=\sum_{n \geqslant 1} \frac{a_{n}}{n^{s}}L(s,f)=∑n⩾1anns
converging absolutely in the range ( s ) > 2 ℜ ( s ) > 2 ℜ(s) > 2\Re(s)>2ℜ(s)>2. Hecke has shown that this can also essentially be expressed as the Mellin transform of the restriction of f f fff to the imaginary line,
(0.3) ( 2 π ) s Γ ( s ) L ( s , f ) = 0 f ( i y ) y s 1 d y (0.3) ( 2 Ï€ ) − s Γ ( s ) L ( s , f ) = ∫ 0 ∞   f ( i y ) y s − 1 d y {:(0.3)(2pi)^(-s)Gamma(s)L(s","f)=int_(0)^(oo)f(iy)y^(s-1)dy:}\begin{equation*} (2 \pi)^{-s} \Gamma(s) L(s, f)=\int_{0}^{\infty} f(i y) y^{s-1} d y \tag{0.3} \end{equation*}(0.3)(2Ï€)−sΓ(s)L(s,f)=∫0∞f(iy)ys−1dy
This formula implies at once two essential analytic properties of L ( s , f ) L ( s , f ) L(s,f)L(s, f)L(s,f) : its analytic continuation to the complex plane and a functional equation of the form s 2 s s ↔ 2 − s s harr2-ss \leftrightarrow 2-ss↔2−s. Moreover, it also has interesting arithmetic content: when specialized to the central value s = 1 s = 1 s=1s=1s=1 and combined with a theorem of Drinfeld and Manin, it allows showing that the ratio between the central value of the L L LLL-function of a (modular) elliptic curve and its (unique) real period is always rational as predicted by a refinement of the Birch-Swinnerton-Dyer conjecture.
The above formula of Hecke can be reformulated (and slightly generalized) in the language of adelic groups and automorphic forms as follows. Let π Ï€ pi\piÏ€ be a cuspidal automorphic representation of PGL 2 ( A ) PGL 2 ⁡ ( A ) PGL_(2)(A)\operatorname{PGL}_{2}(\mathbb{A})PGL2⁡(A), where A = R × p Q p A = R × ∏ p ′   Q p A=Rxxprod_(p)^(')Q_(p)\mathbb{A}=\mathbb{R} \times \prod_{p}^{\prime} \mathbb{Q}_{p}A=R×∏p′Qp denotes the adele ring of the rationals. This roughly means that π Ï€ pi\piÏ€ is an irreducible representation realized in a space of smooth and rapidly decreasing functions on P G L 2 ( Q ) P G L 2 ( A ) P G L 2 ( Q ) ∖ P G L 2 ( A ) PGL_(2)(Q)\\PGL_(2)(A)\mathrm{PGL}_{2}(\mathbb{Q}) \backslash \mathrm{PGL}_{2}(\mathbb{A})PGL2(Q)∖PGL2(A). Then, for every φ π φ ∈ Ï€ varphi in pi\varphi \in \piφ∈π we have an identity of the following shape:
(0.4) A ( Q ) A ( A ) φ ( a ) d a L ( 1 2 , π ) (0.4) ∫ A ( Q ) ∖ A ( A )   φ ( a ) d a ∼ L 1 2 , Ï€ {:(0.4)int_(A(Q)\\A(A))varphi(a)da∼L((1)/(2),pi):}\begin{equation*} \int_{A(\mathbb{Q}) \backslash A(\mathbb{A})} \varphi(a) d a \sim L\left(\frac{1}{2}, \pi\right) \tag{0.4} \end{equation*}(0.4)∫A(Q)∖A(A)φ(a)da∼L(12,Ï€)
where A = ( ) A = ⋆ ⋆ A=(^(***)_(***))A=\left({ }^{\star}{ }_{\star}\right)A=(⋆⋆) is the standard split torus in P G L 2 P G L 2 PGL_(2)\mathrm{PGL}_{2}PGL2 and L ( s , π ) L ( s , Ï€ ) L(s,pi)L(s, \pi)L(s,Ï€) is the L L LLL-function of π Ï€ pi\piÏ€, a particular instance of the notion of automorphic L L LLL-functions defined by Langlands. For specific π Ï€ pi\piÏ€ 's, this recovers Hecke's formula (0.3) for s = 1 s = 1 s=1s=1s=1, although L ( s , π ) L ( s , Ï€ ) L(s,pi)L(s, \pi)L(s,Ï€) then coincides with the L L LLL-functions of a modular form only up to a renormalization that moves its center of symmetry to 1 / 2 1 / 2 1//21 / 21/2. Moreover, the ∼ ∼\sim∼ sign means that the equality only holds up to other, arguably more elementary, multiplicative factors.
Let E / Q E / Q E//QE / \mathbb{Q}E/Q be a quadratic extension. In the 1980s, Waldspurger [46] has established another striking formula for the central value of the base-change L L LLL-function
L ( s , π E ) = L ( s , π ) L ( s , π χ E ) L s , Ï€ E = L ( s , Ï€ ) L s , Ï€ ⊗ χ E L(s,pi_(E))=L(s,pi)L(s,pi oxchi_(E))L\left(s, \pi_{E}\right)=L(s, \pi) L\left(s, \pi \otimes \chi_{E}\right)L(s,Ï€E)=L(s,Ï€)L(s,π⊗χE)
where χ E : A × / Q × Ï‡ E : A × / Q × chi_(E):A^(xx)//Q^(xx)\chi_{E}: \mathbb{A}^{\times} / \mathbb{Q}^{\times}χE:A×/Q×is the idele class character associated to the extension E / Q E / Q E//QE / \mathbb{Q}E/Q. Waldspurger's formula roughly takes the following shape:
(0.5) | T ( Q ) T ( A ) φ ( t ) d t | 2 L ( 1 2 , π E ) (0.5) ∫ T ( Q ) ∖ T ( A )   φ ( t ) d t 2 ∼ L 1 2 , Ï€ E {:(0.5)|int_(T(Q)\\T(A))varphi(t)dt|^(2)∼L((1)/(2),pi_(E)):}\begin{equation*} \left|\int_{T(\mathbb{Q}) \backslash T(\mathbb{A})} \varphi(t) d t\right|^{2} \sim L\left(\frac{1}{2}, \pi_{E}\right) \tag{0.5} \end{equation*}(0.5)|∫T(Q)∖T(A)φ(t)dt|2∼L(12,Ï€E)
for φ π φ ∈ Ï€ varphi in pi\varphi \in \piφ∈π, where T T TTT is a torus in P G L 2 P G L 2 PGL_(2)\mathrm{PGL}_{2}PGL2 isomorphic to Res E / F ( G m ) / G m ( Res E / F Res E / F ⁡ G m / G m Res E / F Res_(E//F)(G_(m))//G_(m)(Res_(E//F):}\operatorname{Res}_{E / F}\left(\mathbb{G}_{m}\right) / \mathbb{G}_{m}\left(\operatorname{Res}_{E / F}\right.ResE/F⁡(Gm)/Gm(ResE/F denoting Weil's restriction of scalars). This result has lead in the subsequent years to striking arithmetic applications such as to the Birch-Swinnerton-Dyer conjecture or to p p ppp-adic L L LLL-functions.
Although of a similar shape, the two formulas (0.4) and (0.5) also have important differences, e.g., the left-hand side of (0.5) is typically far more algebraic in nature, and indeed sometimes just reduces to a finite sum, whereas the formula (0.4) can be deformed to all complex number s s sss, giving an integral representation of the L L LLL-function L ( s , π ) L ( s , Ï€ ) L(s,pi)L(s, \pi)L(s,Ï€) as Hecke's original formula, and typically carries information that is more transcendental.
The left-hand sides of (0.4) and (0.5) are particular instances of automorphic periods that can be informally defined as the integral of an automorphic form over a subgroup. We can also consider these two period integrals in a more representation-theoretic way as giving explicit A ( A ) A ( A ) A(A)A(\mathbb{A})A(A) - or T ( A ) T ( A ) T(A)T(\mathbb{A})T(A)-invariant linear forms on π Ï€ pi\piÏ€. This point of view rapidly leads to a local
related problem which, given a place v v vvv of Q Q Q\mathbb{Q}Q, aims to describe the irreducible representations of P G L 2 ( Q v ) P G L 2 Q v PGL_(2)(Q_(v))\mathrm{PGL}_{2}\left(\mathbb{Q}_{v}\right)PGL2(Qv) admitting a nonzero A ( Q v ) A Q v A(Q_(v))A\left(\mathbb{Q}_{v}\right)A(Qv) - or T ( Q v ) T Q v T(Q_(v))T\left(\mathbb{Q}_{v}\right)T(Qv)-invariant linear form. It turns out that for the torus A A AAA the answer is always positive except for some degenerate one-dimensional representations. On the other hand, the answer for the torus T T TTT is far more subtle and involves local ε ε epsi\varepsilonε-factors as shown by Tunnell and Saito [44].
A natural generalization of Hecke's formula (0.4) is given by the theory of so-called Rankin-Selberg convolutions as developed by Jacquet Piatetski-Shapiro and Shalika [31]. On the other hand, the Gan-Gross-Prasad conjectures [23] aim to give far-reaching higher-rank generalizations of the above result of Waldspurger as well as of the theorem of Tunnell-Saito.
There has been a lot of progress on these conjectures, as well as some refinements thereof, in recent years, in particular in the case of unitary groups. In this paper, we will survey some of these developments with a particular emphasis on the use of (various forms of) relative trace formulae. Actually, a point I will try to advocate here is that the long journey towards the Gan-Gross-Prasad conjectures was also the occasion to develop and discover new features of these trace formulae.
The content is roughly divided as follows. In the first section, we review the local conjectures of Gan-Gross-Prasad and discuss their proofs in some cases based on some local trace formulae. Then, in Section 2, we introduce the global conjectures for unitary groups, as well as their refinements by Ichino-Ikeda, and describe an approach to both of them through a comparison of global relative trace formulae proposed by Jacquet and Rallis. The next two sections, Sections 3 and 4 , are devoted to explaining the various ingredients needed to carry out this comparison effectively. In the final Section 5, we offer few thoughts about possible future developments.

1. THE LOCAL CONJECTURES AND MULTIPLICITY FORMULAE

1.1. The branching problem

Let F F FFF be a local field (of any characteristic) and E E EEE be either a separable quadratic extension of F F FFF or F F FFF itself. In the case where [ E : F ] = 2 [ E : F ] = 2 [E:F]=2[E: F]=2[E:F]=2, we let c c ccc denote the nontrivial F F FFF-automorphism of E E EEE and otherwise, to obtain uniform notation, we simply set c = 1 c = 1 c=1c=1c=1. Let V V VVV be a Hermitian or quadratic space over E E EEE i.e. a finite dimensional E E EEE-vector space equipped with a nondegenerate c c ccc-sesquilinear form
h : V × V E h : V × V → E h:V xx V rarr Eh: V \times V \rightarrow Eh:V×V→E
satisfying h ( v , w ) = h ( w , v ) c h ( v , w ) = h ( w , v ) c h(v,w)=h(w,v)^(c)h(v, w)=h(w, v)^{c}h(v,w)=h(w,v)c for every v , w V v , w ∈ V v,w in Vv, w \in Vv,w∈V. Let W V W ⊂ V W sub VW \subset VW⊂V be a nondegenerate subspace and let U ( V ) U ( V ) U(V)U(V)U(V) (resp. U ( W ) U ( W ) U(W)U(W)U(W) ) be the group of E E EEE-linear automorphisms g G L E ( V ) g ∈ G L E ( V ) g inGL_(E)(V)g \in \mathrm{GL}_{E}(V)g∈GLE(V) (resp. g G L E ( W ) g ∈ G L E ( W ) g inGL_(E)(W)g \in \mathrm{GL}_{E}(W)g∈GLE(W) ) that preserve the form h h hhh and are of determinant one when E = F E = F E=FE=FE=F. In other words, U ( V ) , U ( W ) U ( V ) , U ( W ) U(V),U(W)U(V), U(W)U(V),U(W) are the unitary groups associated of the Hermitian spaces V , W V , W V,WV, WV,W when [ E : F ] = 2 [ E : F ] = 2 [E:F]=2[E: F]=2[E:F]=2 and the special orthogonal groups of the quadratic spaces V , W V , W V,WV, WV,W when E = F E = F E=FE=FE=F. Note that there is a natural embedding U ( W ) U ( V ) U ( W ) ↪ U ( V ) U(W)↪U(V)U(W) \hookrightarrow U(V)U(W)↪U(V) given by extending the action of g U ( W ) g ∈ U ( W ) g in U(W)g \in U(W)g∈U(W) trivially on the orthogonal complement Z = W Z = W ⊥ Z=W^(_|_)Z=W^{\perp}Z=W⊥ of W W WWW in V V VVV. We assume that
(1.1) Z is odd-dimensional and U ( Z ) is quasisplit. (1.1) Z  is odd-dimensional and  U ( Z )  is quasisplit.  {:(1.1)Z" is odd-dimensional and "U(Z)" is quasisplit. ":}\begin{equation*} Z \text { is odd-dimensional and } U(Z) \text { is quasisplit. } \tag{1.1} \end{equation*}(1.1)Z is odd-dimensional and U(Z) is quasisplit. 
Concretely, this means that there exists a basis ( z i ) r i r z i − r ⩽ i ⩽ r (z_(i))_(-r <= i <= r)\left(z_{i}\right)_{-r \leqslant i \leqslant r}(zi)−r⩽i⩽r of Z Z ZZZ and ν F × Î½ ∈ F × nu inF^(xx)\nu \in F^{\times}ν∈F×such that h ( z i , z j ) = ν δ i , j h z i , z j = ν δ i , − j h(z_(i),z_(j))=nudelta_(i,-j)h\left(z_{i}, z_{j}\right)=\nu \delta_{i,-j}h(zi,zj)=νδi,−j for r i , j r − r ⩽ i , j ⩽ r -r <= i,j <= r-r \leqslant i, j \leqslant r−r⩽i,j⩽r. Let N U ( V ) N ⊂ U ( V ) N sub U(V)N \subset U(V)N⊂U(V) be the unipotent radical of a parabolic subgroup P U ( V ) P ⊂ U ( V ) P sub U(V)P \subset U(V)P⊂U(V) stabilizing a maximal flag of isotropic subspaces in Z Z ZZZ, e.g., with a basis as before, we can take the flag E z r E z r E z r 1 E z r E z 1 E z r ⊂ E z r ⊕ E z r − 1 ⊂ ⋯ ⊂ E z r ⊕ ⋯ ⊕ E z 1 Ez_(r)sub Ez_(r)o+Ez_(r-1)sub cdots sub Ez_(r)o+cdots o+Ez_(1)E z_{r} \subset E z_{r} \oplus E z_{r-1} \subset \cdots \subset E z_{r} \oplus \cdots \oplus E z_{1}Ezr⊂Ezr⊕Ezr−1⊂⋯⊂Ezr⊕⋯⊕Ez1. Then, U ( W ) U ( W ) U(W)U(W)U(W) normalizes N N NNN and Gan-Gross-Prasad construct a certain conjugacy class of U ( W ) U ( W ) U(W)U(W)U(W)-invariant characters ξ : N C × Î¾ : N → C × xi:N rarrC^(xx)\xi: N \rightarrow \mathbb{C}^{\times}ξ:N→C×. Concretely, we can take
ξ ( u ) = ψ ( i = 0 r 1 h ( u z i , z i 1 ) ) , u N ξ ( u ) = ψ ∑ i = 0 r − 1   h u z i , z − i − 1 , u ∈ N xi(u)=psi(sum_(i=0)^(r-1)h(uz_(i),z_(-i-1))),quad u in N\xi(u)=\psi\left(\sum_{i=0}^{r-1} h\left(u z_{i}, z_{-i-1}\right)\right), \quad u \in Nξ(u)=ψ(∑i=0r−1h(uzi,z−i−1)),u∈N
where ψ : F C × Ïˆ : F → C × psi:F rarrC^(xx)\psi: F \rightarrow \mathbb{C}^{\times}ψ:F→C×is a nontrivial character.
The local GGP conjectures roughly address the following branching problems: for smooth irreducible complex representations ( π , V π ) Ï€ , V Ï€ (pi,V_(pi))\left(\pi, V_{\pi}\right)(Ï€,VÏ€) and ( σ , V σ ) σ , V σ (sigma,V_(sigma))\left(\sigma, V_{\sigma}\right)(σ,Vσ) of U ( V ) U ( V ) U(V)U(V)U(V) and U ( W ) U ( W ) U(W)U(W)U(W) respectively, determine the dimension (also called multiplicity) of the following intertwining space:
(1.2) m ( π , σ ) = dim Hom U ( W ) N ( V π , V σ ξ ) (1.2) m ( Ï€ , σ ) = dim ⁡ Hom U ( W ) ∝ N ⁡ V Ï€ , V σ ⊗ ξ {:(1.2)m(pi","sigma)=dim Hom_(U(W)prop N)(V_(pi),V_(sigma)ox xi):}\begin{equation*} m(\pi, \sigma)=\operatorname{dim} \operatorname{Hom}_{U(W) \propto N}\left(V_{\pi}, V_{\sigma} \otimes \xi\right) \tag{1.2} \end{equation*}(1.2)m(Ï€,σ)=dim⁡HomU(W)∝N⁡(VÏ€,Vσ⊗ξ)
Here, when F F FFF is Archimedean by a smooth representation we actually mean an admissible smooth Fréchet representation of moderate growth in the sense of Casselman-Wallach [19]. Moreover, in this case V π , V σ V Ï€ , V σ V_(pi),V_(sigma)V_{\pi}, V_{\sigma}VÏ€,Vσ are Fréchet spaces and by definition Hom U ( W ) N ( V π , V σ ξ ) Hom U ( W ) ⋉ N ⁡ V Ï€ , V σ ⊗ ξ Hom_(U(W)|><N)(V_(pi),V_(sigma)ox xi)\operatorname{Hom}_{U(W) \ltimes N}\left(V_{\pi}, V_{\sigma} \otimes \xi\right)HomU(W)⋉N⁡(VÏ€,Vσ⊗ξ) only consists in the continuous U ( W ) N U ( W ) ⋉ N U(W)|><NU(W) \ltimes NU(W)⋉N-equivariant intertwining maps.
By deep theorems of Aizenbud-Gourevitch-Rallis-Schiffmann [2] in the p p ppp-adic case and Sun-Zhu [42] in the Archimedean case, the branching multiplicity m ( π , σ ) m ( Ï€ , σ ) m(pi,sigma)m(\pi, \sigma)m(Ï€,σ) is known to always be at most 1 (at least when F F FFF is of characteristic 0 , see [37] for the case of positive characteristic). Thus, the question reduces to determine when m ( π , σ ) m ( Ï€ , σ ) m(pi,sigma)m(\pi, \sigma)m(Ï€,σ) is nonzero.
Gan, Gross, and Prasad formulated a precise answer to this question, under some restrictions on the representations π Ï€ pi\piÏ€ and σ σ sigma\sigmaσ, based on the so-called Langlands correspondences for the groups U ( V ) U ( V ) U(V)U(V)U(V) and U ( W ) U ( W ) U(W)U(W)U(W). More precisely, these give ways to parametrize smooth irreducible representations of those groups in terms of L L LLL-parameters which are certain kind of morphisms
ϕ : L F L U ( V ) or L U ( W ) Ï• : L F → L U ( V )  or  L U ( W ) phi:L_(F)rarr^(L)U(V)quad" or "quad^(L)U(W)\phi: \mathscr{L}_{F} \rightarrow{ }^{L} U(V) \quad \text { or } \quad{ }^{L} U(W)Ï•:LF→LU(V) or LU(W)
from a group L F L F L_(F)\mathscr{L}_{F}LF which can be taken to be either the Weil group W F W F W_(F)W_{F}WF (in the Archimedean case) or a product W F × S L 2 ( C W F × S L 2 ( C W_(F)xxSL_(2)(CW_{F} \times \mathrm{SL}_{2}(\mathbb{C}WF×SL2(C ) (in the non-Archimedean case) to a semidirect product L U ( V ) = U ( V ) ^ W F L U ( V ) = U ( V ) ^ ⋊ W F ^(L)U(V)= widehat(U(V))><|W_(F){ }^{L} U(V)=\widehat{U(V)} \rtimes W_{F}LU(V)=U(V)^⋊WF or L U ( W ) = U ( W ) ^ W F L U ( W ) = U ( W ) ^ ⋊ W F ^(L)U(W)= widehat(U(W))><|W_(F){ }^{L} U(W)=\widehat{U(W)} \rtimes W_{F}LU(W)=U(W)^⋊WF known as the L-group. In the cases at hand, the connected components U ( V ) ^ U ( V ) ^ widehat(U(V))\widehat{U(V)}U(V)^ and U ( W ) ^ U ( W ) ^ widehat(U(W))\widehat{U(W)}U(W)^ turn out to be either complex general linear groups (in the unitary case) or complex special orthogonal/symplectic groups (in the orthogonal case) and the relevant sets of L L LLL-parameters can be more concretely described as sets of complex representations of L E L E L_(E)\mathscr{L}_{E}LE of fixed dimension and satisfying certain properties of (conjugate-)self-duality. We refer the reader to [23, §8] for details and content ourself to briefly sketch this alternative description for unitary groups: the L L LLL-parameters for U ( V ) U ( V ) U(V)U(V)U(V) can be equivalently described as isomorphism classes of n = dim ( V ) n = dim ⁡ ( V ) n=dim(V)n=\operatorname{dim}(V)n=dim⁡(V)-dimensional complex semisimple representations ϕ : L E G L ( M ) Ï• : L E → G L ( M ) phi:L_(E)rarrGL(M)\phi: \mathscr{L}_{E} \rightarrow \mathrm{GL}(M)Ï•:LE→GL(M) which are conjugate-self-dual of sign ( 1 ) n 1 ( − 1 ) n − 1 (-1)^(n-1)(-1)^{n-1}(−1)n−1. Here, ϕ Ï• phi\phiÏ• is said to be conjugate-self-dual if there is an isomorphism T : M M σ T : M → M ∨ σ T:M rarrM^(vv sigma)T: M \rightarrow M^{\vee \sigma}T:M→M∨σ
with its conjugate-contragredient ϕ σ : L E G L ( M σ ) Ï• ∨ σ : L E → G L M ∨ σ phi^(vv sigma):L_(E)rarrGL(M^(vv sigma))\phi^{\vee \sigma}: \mathscr{L}_{E} \rightarrow \mathrm{GL}\left(M^{\vee \sigma}\right)ϕ∨σ:LE→GL(M∨σ) obtained by twisting the contragredient by any chosen lift σ L F L E σ ∈ L F ∖ L E sigma inL_(F)\\L_(E)\sigma \in \mathscr{L}_{F} \backslash \mathscr{L}_{E}σ∈LF∖LE of c c ccc and it is, moreover, said to be of sign ε { ± } ε ∈ { ± } epsi in{+-}\varepsilon \in\{ \pm\}ε∈{±} if the isomorphism T T TTT can be chosen so that t T ϕ ( σ 2 ) = ε T t T Ï• σ 2 = ε T ^(t)T phi(sigma^(2))=epsi T{ }^{t} T \phi\left(\sigma^{2}\right)=\varepsilon TtTÏ•(σ2)=εT. Besides these L L LLL-parameters ϕ Ï• phi\phiÏ•, the local Langlands correspondence is also supposed to associate to irreducible representations irreducible characters of the finite group of components
S ϕ = π 0 ( Cent U ( V ) ^ ( ϕ ) ) S Ï• = Ï€ 0 Cent U ( V ) ^ ⁡ ( Ï• ) S_(phi)=pi_(0)(Cent_( widehat(U(V)))(phi))S_{\phi}=\pi_{0}\left(\operatorname{Cent}_{\widehat{U(V)}}(\phi)\right)SÏ•=Ï€0(CentU(V)^⁡(Ï•))
of the centralizer of the image of ϕ Ï• phi\phiÏ• in U ( V ) ^ U ( V ) ^ widehat(U(V))\widehat{U(V)}U(V)^. For the group considered here, S ϕ S Ï• S_(phi)S_{\phi}SÏ• is always a 2-group and moreover, once again, it also admits a more concrete description, e.g., if U ( V ) U ( V ) U(V)U(V)U(V) is a unitary group and we identify ϕ Ï• phi\phiÏ• with a ( 1 ) n 1 ( − 1 ) n − 1 (-1)^(n-1)(-1)^{n-1}(−1)n−1-conjugate-self-dual representation of L E L E L_(E)\mathscr{L}_{E}LE as before, this can be decomposed into irreducible representations as follows:
(1.3) ϕ = i I n i ϕ i j J m j ϕ j k K l k ( ϕ k ϕ k σ ) (1.3) Ï• = ⨁ i ∈ I   n i Ï• i ⨁ j ∈ J   m j Ï• j ⨁ k ∈ K   l k Ï• k ⊕ Ï• k ∨ σ {:(1.3)phi=bigoplus_(i in I)n_(i)phi_(i)bigoplus_(j in J)m_(j)phi_(j)bigoplus_(k in K)l_(k)(phi_(k)o+phi_(k)^(vv sigma)):}\begin{equation*} \phi=\bigoplus_{i \in I} n_{i} \phi_{i} \bigoplus_{j \in J} m_{j} \phi_{j} \bigoplus_{k \in K} l_{k}\left(\phi_{k} \oplus \phi_{k}^{\vee \sigma}\right) \tag{1.3} \end{equation*}(1.3)Ï•=⨁i∈IniÏ•i⨁j∈JmjÏ•j⨁k∈Klk(Ï•k⊕ϕk∨σ)
where the ϕ i Ï• i phi_(i)\phi_{i}Ï•i 's (resp. ϕ j Ï• j phi_(j)\phi_{j}Ï•j 's) are irreducible conjugate-self-dual of the same sign ( 1 ) n 1 ( − 1 ) n − 1 (-1)^(n-1)(-1)^{n-1}(−1)n−1 (resp. of opposite sign ( 1 ) n ( − 1 ) n (-1)^(n)(-1)^{n}(−1)n ) whereas the ϕ k Ï• k phi_(k)\phi_{k}Ï•k 's are irreducible but not conjugate-self-dual and using this decomposition we have
(1.4) S ϕ = i I Z / 2 Z e i (1.4) S Ï• = ⨁ i ∈ I   Z / 2 Z e i {:(1.4)S_(phi)=bigoplus_(i in I)Z//2Ze_(i):}\begin{equation*} S_{\phi}=\bigoplus_{i \in I} \mathbb{Z} / 2 \mathbb{Z} e_{i} \tag{1.4} \end{equation*}(1.4)SÏ•=⨁i∈IZ/2Zei
We are now ready to state a version of the local Langlands correspondence, including an essential refinement by Vogan [45], necessary for the local Gan-Gross-Prasad conjecture. It turns out to be more easily described if we consider more than one group at the same time: besides U ( V ) 1 U ( V ) 1 U(V)^(1)U(V){ }^{1}U(V)1 itself, we need to consider its pure inner forms which here consist of the groups U ( V ) U V ′ U(V^('))U\left(V^{\prime}\right)U(V′) where V V ′ V^(')V^{\prime}V′ runs over the isomorphism classes of Hermitian/quadratic spaces of the same dimension as V V VVV and of the same discriminant in the orthogonal case. If F F FFF is non-Archimedean, and provided V V VVV is not an hyperbolic quadratic plane, there are always two such isomorphism classes of Hermitian/quadratic spaces and thus as many pure inner forms whereas if F F FFF is Archimedean, by their classification using signatures there are dim ( V ) + 1 dim ⁡ ( V ) + 1 dim(V)+1\operatorname{dim}(V)+1dim⁡(V)+1 (resp. dim ( V ) + 1 2 dim ⁡ ( V ) + 1 2 (dim(V)+1)/(2)\frac{\operatorname{dim}(V)+1}{2}dim⁡(V)+12 for dim ( V ) dim ⁡ ( V ) dim(V)\operatorname{dim}(V)dim⁡(V) odd, dim ( V ) + disc ( V ) + 1 2 dim ⁡ ( V ) + disc ⁡ ( V ) + 1 2 (dim(V)+disc(V)+1)/(2)\frac{\operatorname{dim}(V)+\operatorname{disc}(V)+1}{2}dim⁡(V)+disc⁡(V)+12 for dim ( V ) dim ⁡ ( V ) dim(V)\operatorname{dim}(V)dim⁡(V) even) pure inner forms in the unitary case (resp. orthogonal case). For such a pure inner form, let us denote by Irr ( U ( V ) ) Irr ⁡ U V ′ Irr(U(V^(')))\operatorname{Irr}\left(U\left(V^{\prime}\right)\right)Irr⁡(U(V′)) the set of isomorphism classes of smooth irreducible representations of U ( V ) U V ′ U(V^('))U\left(V^{\prime}\right)U(V′). Then, modulo the auxilliary choice of a quasisplit pure inner form U ( V ) U V ′ U(V^('))U\left(V^{\prime}\right)U(V′) and a Whittaker datum on i t 2 i t 2 it^(2)\mathrm{it}^{2}it2 that we will suppress from the notation, the local Langlands correspondence posits the existence of a natural decomposition into finite sets called L L LLL-packets
V Irr ( U ( V ) ) = ϕ Π ( ϕ ) ⨆ V ′   Irr ⁡ U V ′ = ⨆ Ï•   Π ( Ï• ) ⨆_(V^('))Irr(U(V^(')))=⨆_(phi)Pi(phi)\bigsqcup_{V^{\prime}} \operatorname{Irr}\left(U\left(V^{\prime}\right)\right)=\bigsqcup_{\phi} \Pi(\phi)⨆V′Irr⁡(U(V′))=⨆ϕΠ(Ï•)
where the left union runs over all pure inner forms whereas the right union is over all L L LLL-parameters ϕ : L F L U ( V ) Ï• : L F → L U ( V ) phi:L_(F)rarr^(L)U(V)\phi: \mathscr{L}_{F} \rightarrow{ }^{L} U(V)Ï•:LF→LU(V) (the pure inner forms all share the same L L LLL-group) together with bijections
(1.5) Π ( ϕ ) S ϕ ^ π ( ϕ , χ ) χ (1.5) Π ( Ï• ) ≃ S Ï• ^ Ï€ ( Ï• , χ ) ← χ {:[(1.5)Pi(phi)≃ widehat(S_(phi))],[pi(phi","chi)larr chi]:}\begin{align*} & \Pi(\phi) \simeq \widehat{S_{\phi}} \tag{1.5}\\ & \pi(\phi, \chi) \leftarrow \chi \end{align*}(1.5)Π(Ï•)≃SÏ•^Ï€(Ï•,χ)←χ
with the character group S ϕ ^ S Ï• ^ widehat(S_(phi))\widehat{S_{\phi}}SÏ•^ of S ϕ S Ï• S_(phi)S_{\phi}SÏ•. Thus, in a sense the correspondence gives a way to parameterize the admissible duals of all the pure inner forms of U ( V ) U ( V ) U(V)U(V)U(V) at the same time.

tion Π V ( ϕ ) = Π ( ϕ ) Irr ( U ( V ) ) Π V ′ ( Ï• ) = Π ( Ï• ) ∩ Irr ⁡ U V ′ Pi^(V^('))(phi)=Pi(phi)nn Irr(U(V^(')))\Pi^{V^{\prime}}(\phi)=\Pi(\phi) \cap \operatorname{Irr}\left(U\left(V^{\prime}\right)\right)ΠV′(Ï•)=Π(Ï•)∩Irr⁡(U(V′)) and therefore this also induces a parameterization of the individual admissible duals Irr ( U ( V ) Irr ⁡ U V ′ Irr(U(V^(')):}\operatorname{Irr}\left(U\left(V^{\prime}\right)\right.Irr⁡(U(V′) ). Moreover, the naturality condition can be made precise through the so-called endoscopic relations that characterize the Langlands parameterization uniquely in terms of the known correspondence for G L n . 3 G L n . 3 GL_(n).^(3)\mathrm{GL}_{n} .{ }^{3}GLn.3 For real groups, the correspondence was constructed long ago by Langlands and is known to satisfy the endoscopic relations thanks to the work of Shelstad. In his monumental work [7], Arthur has established, among other things, the existence of this correspondence for quasisplit special orthogonal or symplectic p p ppp-adic groups (with an important technical caveat for even special orthogonal groups S O ( 2 n ) S O ( 2 n ) SO(2n)\mathrm{SO}(2 n)SO(2n) where the correspondence is only proven up to conjugation by the full orthogonal group O ( 2 n ) O ( 2 n ) O(2n)O(2 n)O(2n) ). This work was subsequently extended in [39] and [34] to include unitary groups (not necessarily quasisplit).
For the purpose of stating the local Gan-Gross-Prasad conjecture, we will also need to vary the two groups U ( V ) , U ( W ) U ( V ) , U ( W ) U(V),U(W)U(V), U(W)U(V),U(W). However, we will need these to vary in a compatible way in order for the multiplicity (1.2) to still be well-defined. More precisely, the relevant pure inner forms of U ( V ) × U ( W ) U ( V ) × U ( W ) U(V)xx U(W)U(V) \times U(W)U(V)×U(W) are defined by varying the small Hermitian/quadratic space W W WWW while keeping the orthogonal complement Z = W Z = W ⊥ Z=W^(_|_)Z=W^{\perp}Z=W⊥ fixed: these are the groups of the form U ( V ) × U ( W ) U V ′ × U W ′ U(V^('))xx U(W^('))U\left(V^{\prime}\right) \times U\left(W^{\prime}\right)U(V′)×U(W′) where W W ′ W^(')W^{\prime}W′ is a Hermitian/quadratic space of the same dimension as W W WWW, and same discriminant in the orthogonal case, whereas V V ′ V^(')V^{\prime}V′ is given by the orthogonal sum V = W Z V ′ = W ′ ⊕ ⊥ Z V^(')=W^(')o+^(_|_)ZV^{\prime}=W^{\prime} \oplus^{\perp} ZV′=W′⊕⊥Z. Since the orthogonal complement Z Z ZZZ is the same, for each relevant pure inner form U ( V ) × U ( W ) U V ′ × U W ′ U(V^('))xx U(W^('))U\left(V^{\prime}\right) \times U\left(W^{\prime}\right)U(V′)×U(W′) we can define as before a multiplicity function ( π , σ ) ( Ï€ , σ ) ∈ (pi,sigma)in(\pi, \sigma) \in(Ï€,σ)∈ Irr ( U ( V ) ) × Irr ( U ( W ) ) m ( π , σ ) Irr ⁡ U V ′ × Irr ⁡ U W ′ ↦ m ( Ï€ , σ ) Irr(U(V^(')))xx Irr(U(W^(')))|->m(pi,sigma)\operatorname{Irr}\left(U\left(V^{\prime}\right)\right) \times \operatorname{Irr}\left(U\left(W^{\prime}\right)\right) \mapsto m(\pi, \sigma)Irr⁡(U(V′))×Irr⁡(U(W′))↦m(Ï€,σ).
We are now ready to formulate the local Gan-Gross-Prasad conjecture except for one technical but important detail: as alluded to above, the local Langlands correspondences, and more particularly the bijections (1.5), depend on the choice of Whittaker data on some pure inner forms of U ( V ) U ( V ) U(V)U(V)U(V) and U ( W ) U ( W ) U(W)U(W)U(W). Actually, it turns out that there exists a unique relevant pure inner form U ( V q s ) × U ( W q s ) U V q s × U W q s U(V_(qs))xx U(W_(qs))U\left(V_{q s}\right) \times U\left(W_{q s}\right)U(Vqs)×U(Wqs) which is quasisplit and on which we can fix a Whittaker datum through the choice of a nontrivial character ψ : E C × Ïˆ : E → C × psi:E rarrC^(xx)\psi: E \rightarrow \mathbb{C}^{\times}ψ:E→C×that is, moreover, trivial for F F FFF in the unitary case (see [23, $12] for details). With these prerequisites in place, we can now state:
Conjecture 1.1 (Gan-Gross-Prasad). Let ϕ : L F L U ( V ) Ï• : L F → L U ( V ) phi:L_(F)rarr^(L)U(V)\phi: \mathscr{L}_{F} \rightarrow{ }^{L} U(V)Ï•:LF→LU(V) and ϕ : L F L U ( W ) Ï• ′ : L F → L U ( W ) phi^('):L_(F)rarr^(L)U(W)\phi^{\prime}: \mathscr{L}_{F} \rightarrow{ }^{L} U(W)ϕ′:LF→LU(W) be L-parameters for U ( V ) U ( V ) U(V)U(V)U(V) and U ( W ) U ( W ) U(W)U(W)U(W), respectively. Assume that the corresponding L-packets
3 This situation is peculiar to classical groups because those can be realized as twisted endoscopic groups of some G L N G L N GL_(N)\mathrm{GL}_{N}GLN.
Π ( ϕ ) , Π ( ϕ ) Π ( Ï• ) , Π Ï• ′ Pi(phi),Pi(phi^('))\Pi(\phi), \Pi\left(\phi^{\prime}\right)Π(Ï•),Π(ϕ′) are generic, that is, they contain one representation which is generic with respect to each Whittaker datum. Then:
(1) There exists a unique pair
( π , σ ) W Π V ( ϕ ) × Π W ( ϕ ) ( Ï€ , σ ) ∈ ⨆ W ′   Π V ′ ( Ï• ) × Π W ′ Ï• ′ (pi,sigma)in⨆_(W^('))Pi^(V^('))(phi)xxPi^(W^('))(phi^('))(\pi, \sigma) \in \bigsqcup_{W^{\prime}} \Pi^{V^{\prime}}(\phi) \times \Pi^{W^{\prime}}\left(\phi^{\prime}\right)(Ï€,σ)∈⨆W′ΠV′(Ï•)×ΠW′(ϕ′)
where the union runs over relevant pure inner forms, such that m ( π , σ ) = 1 m ( Ï€ , σ ) = 1 m(pi,sigma)=1m(\pi, \sigma)=1m(Ï€,σ)=1.
(2) The unique characters χ S ϕ ^ χ ∈ S Ï• ^ chi in widehat(S_(phi))\chi \in \widehat{S_{\phi}}χ∈SÏ•^ and χ S ϕ ^ χ ′ ∈ S Ï• ′ ^ chi^(')in widehat(S_(phi^(')))\chi^{\prime} \in \widehat{S_{\phi^{\prime}}}χ′∈Sϕ′^ such that π = π ( ϕ , χ ) Ï€ = Ï€ ( Ï• , χ ) pi=pi(phi,chi)\pi=\pi(\phi, \chi)Ï€=Ï€(Ï•,χ) and σ = π ( ϕ , χ ) σ = Ï€ Ï• ′ , χ ′ sigma=pi(phi^('),chi^('))\sigma=\pi\left(\phi^{\prime}, \chi^{\prime}\right)σ=Ï€(ϕ′,χ′) are given by explicit formulas involving local root numbers, e.g., in the unitary case, identifying ϕ , ϕ Ï• , Ï• ′ phi,phi^(')\phi, \phi^{\prime}Ï•,ϕ′ with conjugate-self-dual representations of L E L E L_(E)\mathscr{L}_{E}LE and using the description (1.4) of S ϕ S Ï• ′ S_(phi^('))S_{\phi^{\prime}}Sϕ′ in terms of the decomposition (1.3), we have
(1.6) χ ( e i ) = ε ( ϕ i ϕ , ψ 2 δ ) , for all i I (1.6) χ e i = ε Ï• i ⊗ Ï• ′ , ψ 2 δ ,  for all  i ∈ I {:(1.6)chi(e_(i))=epsi(phi_(i)oxphi^('),psi_(2delta))","quad" for all "i in I:}\begin{equation*} \chi\left(e_{i}\right)=\varepsilon\left(\phi_{i} \otimes \phi^{\prime}, \psi_{2 \delta}\right), \quad \text { for all } i \in I \tag{1.6} \end{equation*}(1.6)χ(ei)=ε(Ï•i⊗ϕ′,ψ2δ), for all i∈I
Here δ δ delta\deltaδ stands for the discriminant of the odd dimensional Hermitian space among ( V q s , W q s ) , ψ 2 δ ( z ) := ψ ( 2 δ z ) V q s , W q s , ψ 2 δ ( z ) := ψ ( 2 δ z ) (V_(qs),W_(qs)),psi_(2delta)(z):=psi(2delta z)\left(V_{q s}, W_{q s}\right), \psi_{2 \delta}(z):=\psi(2 \delta z)(Vqs,Wqs),ψ2δ(z):=ψ(2δz) and ε ( ϕ i ϕ , ψ 2 δ ) ε Ï• i ⊗ Ï• ′ , ψ 2 δ epsi(phi_(i)oxphi^('),psi_(2delta))\varepsilon\left(\phi_{i} \otimes \phi^{\prime}, \psi_{2 \delta}\right)ε(Ï•i⊗ϕ′,ψ2δ) denotes the local root number of the Weil or Weil-Deligne representation ϕ i ϕ Ï• i ⊗ Ï• ′ phi_(i)oxphi^(')\phi_{i} \otimes \phi^{\prime}Ï•i⊗ϕ′ associated to this additive character [43].
When ( dim ( V ) , dim ( W ) ) = ( 3 , 2 ) ( dim ⁡ ( V ) , dim ⁡ ( W ) ) = ( 3 , 2 ) (dim(V),dim(W))=(3,2)(\operatorname{dim}(V), \operatorname{dim}(W))=(3,2)(dim⁡(V),dim⁡(W))=(3,2) (quadratic case) or ( dim ( V ) , dim ( W ) ) = ( 2 , 1 ) ( dim ⁡ ( V ) , dim ⁡ ( W ) ) = ( 2 , 1 ) (dim(V),dim(W))=(2,1)(\operatorname{dim}(V), \operatorname{dim}(W))=(2,1)(dim⁡(V),dim⁡(W))=(2,1) (Hermitian case), the above conjecture essentially reduces to the result of Tunnell and Saito [44] on restrictions of irreducible representations of GL(2) to a maximal torus mentioned in the introduction. There has been a lot of recent progress towards Conjecture 1.1 and here is the status of what is currently known in the characteristic zero case:
Theorem 1.1. Assume that F F FFF is of characteristic 0 . Then:
(1) Both (1) and (2) of Conjecture 1.1 hold true in the following cases: if V , W V , W V,WV, WV,W are Hermitian spaces (i.e., in the unitary case) or if these are quadratic spaces and F F FFF is p p ppp-adic.
(2) Conjecture 1.1 (1) is verified when V V VVV, W W WWW are quadratic spaces and F F FFF is Archimedean.
The first real breakthrough on Conjecture 1.1 was made by Waldspurger who established in a stunning series of papers [38,47-49], the last one in collaboration with MÅ“glin, the full conjecture for p p ppp-adic special orthogonal groups under the assumption that the local Langlands correspondence is known for those groups and have expected properties. In my P h D P h D PhD\mathrm{PhD}PhD thesis [8-10], I extended the method to deal with p p ppp-adic unitary groups therefore obtaining the conjecture under the slightly weaker assumption that the parameters ϕ , ϕ Ï• , Ï• ′ phi,phi^(')\phi, \phi^{\prime}Ï•,ϕ′ are tempered which means that the corresponding L L LLL-packets consist of tempered representations. The extension to generic L L LLL-packets was carried out in the appendix to [24] using crucially a result of Heiermann. Later, I revisited Waldspurger's method which is based on a novel sort of local trace formulae, putting it on firmer grounds, and in the monograph [12] I established
part (1) of the conjecture (sometimes called the multiplicity one property for L-packets) for unitary groups over arbitrary fields of characteristic 0 , thus reproving part of my thesis in the p p ppp-adic case, and still under the assumption that L L LLL-parameters are tempered which as we will see is quite natural from the method. In the meantime, H. He [28] has developed a different approach to the conjecture based on the local θ θ theta\thetaθ-correspondence and very special features of the representation theory of real unitary groups (in particular, this approach cannot deal with p p ppp-adic groups) which allowed him to prove the full conjecture for those groups whenever ϕ Ï• phi\phiÏ• and ϕ Ï• ′ phi^(')\phi^{\prime}ϕ′ are discrete parameters (a stronger condition than being tempered). Recently, this technique was enhanced by H. Xue [54] who was able to show the conjecture for real unitary groups without any restriction. Finally, in the recent preprint [36] Z. Luo adapted my previous work to deal with real special orthogonal groups proving the multiplicity one property for tempered L L LLL-packets.

1.2. Approach through local trace formulae

Let me give more details on the general structure of the approach taken by Waldspurger which was clarified and then further refined in [12]. It is mainly based on one completely novel ingredient that is a formula expressing the multiplicity m ( π , σ ) m ( Ï€ , σ ) m(pi,sigma)m(\pi, \sigma)m(Ï€,σ) in terms of the Harish-Chandra characters of π Ï€ pi\piÏ€ and σ σ sigma\sigmaσ. To be more specific, we recall a deep result of Harish-Chandra asserting that the distribution-character of a smooth irreducible representation π Ï€ pi\piÏ€, i.e. the distribution f C c ( U ( V ) ) f ∈ C c ∞ ( U ( V ) ) ↦ f inC_(c)^(oo)(U(V))|->f \in C_{c}^{\infty}(U(V)) \mapstof∈Cc∞(U(V))↦ Trace π ( f ) Ï€ ( f ) pi(f)\pi(f)Ï€(f), can be represented by a locally L 1 L 1 L^(1)L^{1}L1 function Θ π Θ Ï€ Theta_(pi)\Theta_{\pi}Θπ known as its Harish-Chandra character. The aforementioned formula gives an identity roughly of the form:
(1.7) m ( π , σ ) = Γ ( V , W ) r e g c π ( x ) c σ ( x 1 ) d x (1.7) m ( Ï€ , σ ) = ∫ Γ ( V , W ) r e g   c Ï€ ( x ) c σ x − 1 d x {:(1.7)m(pi","sigma)=int_(Gamma(V,W))^(reg)c_(pi)(x)c_(sigma)(x^(-1))dx:}\begin{equation*} m(\pi, \sigma)=\int_{\Gamma(V, W)}^{\mathrm{reg}} c_{\pi}(x) c_{\sigma}\left(x^{-1}\right) d x \tag{1.7} \end{equation*}(1.7)m(Ï€,σ)=∫Γ(V,W)regcÏ€(x)cσ(x−1)dx
where Γ ( V , W ) Γ ( V , W ) Gamma(V,W)\Gamma(V, W)Γ(V,W) is a certain set of semisimple conjugacy classes in U ( V ) U ( V ) U(V)U(V)U(V) equipped with some measure d x d x dxd xdx reminiscent of Weyl integration formula (although it is more singular than measures coming from maximal tori, e.g., singular orbits are typically not negligible for d x ) , c π ( x ) d x ) , c Ï€ ( x ) dx),c_(pi)(x)d x), c_{\pi}(x)dx),cÏ€(x) and c σ ( x 1 ) c σ x − 1 c_(sigma)(x^(-1))c_{\sigma}\left(x^{-1}\right)cσ(x−1) are renormalized values for the characters Θ π Θ Ï€ Theta_(pi)\Theta_{\pi}Θπ and Θ σ Θ σ Theta_(sigma)\Theta_{\sigma}Θσ, respectively (although these characters are smooth on open dense subsets of regular semisimple elements, they typically blow up at the singular conjugacy classes in Γ ( V , W ) Γ ( V , W ) Gamma(V,W)\Gamma(V, W)Γ(V,W); the renormalization is based on further results of Harish-Chandra describing the local behavior of characters near singular elements), and finally the "reg" sign indicates that the integral itself has sometimes to be regularized in a certain way (or put another way, it is improperly convergent). Originally, formula (1.7) was only proven to hold for tempered representations but through the process of reducing the general conjecture to the tempered case, it was eventually shown a posteriori to hold for every irreducible representations belonging to generic L L LLL-packets. In the degenerate case where U ( V ) U ( V ) U(V)U(V)U(V) is compact, the right-hand side of the integral formula (1.7) reduces to the L 2 L 2 L^(2)L^{2}L2-scalar product of Θ π | U ( W ) Θ Ï€ U ( W ) Theta_(pi)|_(U(W))\left.\Theta_{\pi}\right|_{U(W)}Θπ|U(W) and Θ σ Θ σ Theta_(sigma)\Theta_{\sigma}Θσ and the formula itself is an easy consequence of the orthogonality relations of characters, but in general the formula looks much more mysterious.
Let us sketch very briefly how we can deduce from formula (1.7) the first part of Conjecture 1.1 for tempered parameters (multiplicity one in tempered L L LLL-packets). The idea,
due to Waldspurger, is to take advantage of inner cancellations in the sum
(1.8) W ( π , σ ) m ( π , σ ) = W ( π , σ ) Π V ( ϕ ) × Π W ( ϕ ) Γ ( V , W ) r e g c π ( x ) c σ ( x 1 ) d x (1.8) ∑ W ′   ∑ ( Ï€ , σ )   m ( Ï€ , σ ) = ∑ W ′   ∑ ( Ï€ , σ ) ∈ Π V ′ ( Ï• ) × Π W ′ Ï• ′   ∫ Γ V ′ , W ′ r e g   c Ï€ ( x ) c σ x − 1 d x {:(1.8)sum_(W^('))sum_((pi,sigma))m(pi","sigma)=sum_(W^('))sum_((pi,sigma)inPi^(V^('))(phi)xxPi^(W^('))(phi^(')))int_(Gamma(V^('),W^(')))^(reg)c_(pi)(x)c_(sigma)(x^(-1))dx:}\begin{equation*} \sum_{W^{\prime}} \sum_{(\pi, \sigma)} m(\pi, \sigma)=\sum_{W^{\prime}} \sum_{(\pi, \sigma) \in \Pi^{V^{\prime}}(\phi) \times \Pi^{W^{\prime}}\left(\phi^{\prime}\right)} \int_{\Gamma\left(V^{\prime}, W^{\prime}\right)}^{\mathrm{reg}} c_{\pi}(x) c_{\sigma}\left(x^{-1}\right) d x \tag{1.8} \end{equation*}(1.8)∑W′∑(Ï€,σ)m(Ï€,σ)=∑W′∑(Ï€,σ)∈ΠV′(Ï•)×ΠW′(ϕ′)∫Γ(V′,W′)regcÏ€(x)cσ(x−1)dx
that can be deduced from certain character relations (which are basic instances of the already mentioned endoscopic relations). The first step is to rewrite the sum as
(1.9) W Γ ( V , W ) r e g c ϕ V ( x ) c ϕ W ( x 1 ) d x (1.9) ∑ W ′   ∫ Γ V ′ , W ′ r e g   c Ï• V ′ ( x ) c Ï• ′ W ′ x − 1 d x {:(1.9)sum_(W^('))int_(Gamma(V^('),W^(')))^(reg)c_(phi)^(V^('))(x)c_(phi^('))^(W^('))(x^(-1))dx:}\begin{equation*} \sum_{W^{\prime}} \int_{\Gamma\left(V^{\prime}, W^{\prime}\right)}^{\mathrm{reg}} c_{\phi}^{V^{\prime}}(x) c_{\phi^{\prime}}^{W^{\prime}}\left(x^{-1}\right) d x \tag{1.9} \end{equation*}(1.9)∑W′∫Γ(V′,W′)regcÏ•V′(x)cϕ′W′(x−1)dx
where Θ ϕ V = π Π V ( ϕ ) Θ π , Θ ϕ W = σ Π W ( ϕ ) Θ σ Θ Ï• V ′ = ∑ Ï€ ∈ Π V ′ ( Ï• )   Θ Ï€ , Θ Ï• ′ W ′ = ∑ σ ∈ Π W ′ Ï• ′   Θ σ Theta_(phi)^(V^('))=sum_(pi inPi^(V^('))(phi))Theta_(pi),Theta_(phi^('))^(W^('))=sum_(sigma inPi^(W^('))(phi^(')))Theta_(sigma)\Theta_{\phi}^{V^{\prime}}=\sum_{\pi \in \Pi^{V^{\prime}}(\phi)} \Theta_{\pi}, \Theta_{\phi^{\prime}}^{W^{\prime}}=\sum_{\sigma \in \Pi^{W^{\prime}}\left(\phi^{\prime}\right)} \Theta_{\sigma}ΘϕV′=∑π∈ΠV′(Ï•)Θπ,Θϕ′W′=∑σ∈ΠW′(ϕ′)Θσ and c ϕ V ( x ) , c ϕ W ( x 1 ) c Ï• V ′ ( x ) , c Ï• ′ W ′ x − 1 c_(phi)^(V^('))(x),c_(phi^('))^(W^('))(x^(-1))c_{\phi}^{V^{\prime}}(x), c_{\phi^{\prime}}^{W^{\prime}}\left(x^{-1}\right)cÏ•V′(x),cϕ′W′(x−1) are renormalized values for those characters as before. The first property of the Langlands correspondence that we need is that Θ ϕ V , Θ ϕ W Θ Ï• V ′ , Θ Ï• ′ W ′ Theta_(phi)^(V^(')),Theta_(phi^('))^(W^('))\Theta_{\phi}^{V^{\prime}}, \Theta_{\phi^{\prime}}^{W^{\prime}}ΘϕV′,Θϕ′W′ are stable, i.e., are constant on the union of semisimple regular conjugacy classes that become the same over an algebraic closure (a so-called regular stable conjugacy class). It follows from this stability property that the renormalized functions c ϕ V c Ï• V ′ c_(phi)^(V^('))c_{\phi}^{V^{\prime}}cÏ•V′, c ϕ W c Ï• ′ W ′ c_(phi^('))^(W^('))c_{\phi^{\prime}}^{W^{\prime}}cϕ′W′ are also invariant under a suitable extension of stable conjugation for singular elements. Consequently, the sum of multiplicities can be further rewritten as
(1.10) W ( π , σ ) m ( π , σ ) = W Γ ( V , W ) / stab r e g c ϕ V ( y ) c ϕ W ( y 1 ) d y (1.10) ∑ W ′   ∑ ( Ï€ , σ )   m ( Ï€ , σ ) = ∑ W ′   ∫ Γ V ′ , W ′ /  stab  r e g   c Ï• V ′ ( y ) c Ï• ′ W ′ y − 1 d y {:(1.10)sum_(W^('))sum_((pi,sigma))m(pi","sigma)=sum_(W^('))int_(Gamma(V^('),W^('))//" stab ")^(reg)c_(phi)^(V^('))(y)c_(phi^('))^(W^('))(y^(-1))dy:}\begin{equation*} \sum_{W^{\prime}} \sum_{(\pi, \sigma)} m(\pi, \sigma)=\sum_{W^{\prime}} \int_{\Gamma\left(V^{\prime}, W^{\prime}\right) / \text { stab }}^{\mathrm{reg}} c_{\phi}^{V^{\prime}}(y) c_{\phi^{\prime}}^{W^{\prime}}\left(y^{-1}\right) d y \tag{1.10} \end{equation*}(1.10)∑W′∑(Ï€,σ)m(Ï€,σ)=∑W′∫Γ(V′,W′)/ stab regcÏ•V′(y)cϕ′W′(y−1)dy
where Γ ( V , W ) / Γ V ′ , W ′ / Gamma(V^('),W^('))//\Gamma\left(V^{\prime}, W^{\prime}\right) /Γ(V′,W′)/ stab stands for the space of stable conjugacy classes in Γ ( V , W ) Γ V ′ , W ′ Gamma(V^('),W^('))\Gamma\left(V^{\prime}, W^{\prime}\right)Γ(V′,W′). At this point, it is convenient to make the simplifying assumption that F F FFF is p p ppp-adic and W W WWW is not a split quadratic space of dimension 2 ⩽ 2 <= 2\leqslant 2⩽2. Then, there are exactly two relevant pure inner forms U ( V ) × U ( W ) U ( V ) × U ( W ) U(V)xx U(W)U(V) \times U(W)U(V)×U(W) and U ( V ) × U ( W ) U V ′ × U W ′ U(V^('))xx U(W^('))U\left(V^{\prime}\right) \times U\left(W^{\prime}\right)U(V′)×U(W′) with, say, the first one quasisplit. Moreover, the character relations in this case read
Θ ϕ V ( y ) = ε V Θ ϕ V ( y ) ( resp. Θ ϕ W ( y ) = ε W Θ ϕ W ( y ) ) Θ Ï• V ( y ) = ε V Θ Ï• V ′ y ′  resp.  Θ Ï• ′ W ( y ) = ε W Θ Ï• ′ W ′ y ′ Theta_(phi)^(V)(y)=epsi_(V)Theta_(phi)^(V^('))(y^('))quad(" resp. "Theta_(phi^('))^(W)(y)=epsi_(W)Theta_(phi^('))^(W^('))(y^(')))\Theta_{\phi}^{V}(y)=\varepsilon_{V} \Theta_{\phi}^{V^{\prime}}\left(y^{\prime}\right) \quad\left(\text { resp. } \Theta_{\phi^{\prime}}^{W}(y)=\varepsilon_{W} \Theta_{\phi^{\prime}}^{W^{\prime}}\left(y^{\prime}\right)\right)ΘϕV(y)=εVΘϕV′(y′)( resp. Θϕ′W(y)=εWΘϕ′W′(y′))
for certain signs ε V , ε W { ± 1 } ε V , ε W ∈ { ± 1 } epsi_(V),epsi_(W)in{+-1}\varepsilon_{V}, \varepsilon_{W} \in\{ \pm 1\}εV,εW∈{±1} satisfying ε V ε W = 1 ε V ε W = − 1 epsi_(V)epsi_(W)=-1\varepsilon_{V} \varepsilon_{W}=-1εVεW=−1 and for every regular stable conjugacy classes y , y y , y ′ y,y^(')y, y^{\prime}y,y′ in U ( V ) , U ( V ) U ( V ) , U V ′ U(V),U(V^('))U(V), U\left(V^{\prime}\right)U(V),U(V′) (resp. in U ( W ) , U ( W ) U ( W ) , U W ′ U(W),U(W^('))U(W), U\left(W^{\prime}\right)U(W),U(W′) ) that are related by a certain correspondence (which is just an identity of characteristic polynomials except in the even orthogonal case). This correspondence actually naturally extends to give an embedding Γ ( V , W ) / Γ V ′ , W ′ / Gamma(V^('),W^('))//\Gamma\left(V^{\prime}, W^{\prime}\right) /Γ(V′,W′)/ stab Γ ( V , W ) / ↪ Γ ( V , W ) / ↪Gamma(V,W)//\hookrightarrow \Gamma(V, W) /↪Γ(V,W)/ stab, y y y ′ ↦ y y^(')|->yy^{\prime} \mapsto yy′↦y, for which we have
c ϕ V ( y ) c ϕ W ( y ) = c ϕ V ( y ) c ϕ W ( y ) c Ï• V ( y ) c Ï• ′ W ( y ) = − c Ï• V ′ y ′ c Ï• ′ W ′ y ′ c_(phi)^(V)(y)c_(phi^('))^(W)(y)=-c_(phi)^(V^('))(y^('))c_(phi^('))^(W^('))(y^('))c_{\phi}^{V}(y) c_{\phi^{\prime}}^{W}(y)=-c_{\phi}^{V^{\prime}}\left(y^{\prime}\right) c_{\phi^{\prime}}^{W^{\prime}}\left(y^{\prime}\right)cÏ•V(y)cϕ′W(y)=−cÏ•V′(y′)cϕ′W′(y′)
This implies that in the right-hand side of (1.10), all the terms indexed by Γ ( V , W ) / Γ V ′ , W ′ / Gamma(V^('),W^('))//\Gamma\left(V^{\prime}, W^{\prime}\right) /Γ(V′,W′)/ stab can be cancelled with the corresponding terms coming from their images in Γ ( V , W ) / Γ ( V , W ) / Gamma(V,W)//\Gamma(V, W) /Γ(V,W)/ stab. The only remaining contribution, it turns out, is that of the trivial conjugacy class:
(1.11) W ( π , σ ) m ( π , σ ) = c ϕ V ( 1 ) c ϕ W ( 1 ) (1.11) ∑ W ′   ∑ ( Ï€ , σ )   m ( Ï€ , σ ) = c Ï• V ( 1 ) c Ï• ′ W ( 1 ) {:(1.11)sum_(W^('))sum_((pi,sigma))m(pi","sigma)=c_(phi)^(V)(1)c_(phi^('))^(W)(1):}\begin{equation*} \sum_{W^{\prime}} \sum_{(\pi, \sigma)} m(\pi, \sigma)=c_{\phi}^{V}(1) c_{\phi^{\prime}}^{W}(1) \tag{1.11} \end{equation*}(1.11)∑W′∑(Ï€,σ)m(Ï€,σ)=cÏ•V(1)cϕ′W(1)
which, by a result of Rodier, can be interpreted as the number of representations in the packet Π V ( ϕ ) Π W ( ϕ ) Π V ( Ï• ) ⊗ Π W Ï• ′ Pi^(V)(phi)oxPi^(W)(phi^('))\Pi^{V}(\phi) \otimes \Pi^{W}\left(\phi^{\prime}\right)ΠV(Ï•)⊗ΠW(ϕ′) that are generic with respect to a certain Whittaker datum (actually really an average of such numbers over all Whittaker data in the unitary case). By a third property of tempered L L LLL-packets (existence and unicity of a generic representation for a given Whittaker datum), this number is just 1 and this immediately implies the first part of Conjecture 1.1.
The proof of the multiplicity formula (1.7), on the other hand, is much more involved. Set G = U ( W ) × U ( V ) G = U ( W ) × U ( V ) G=U(W)xx U(V)G=U(W) \times U(V)G=U(W)×U(V) and H = U ( W ) N H = U ( W ) ⋉ N H=U(W)|><NH=U(W) \ltimes NH=U(W)⋉N that we see as a subgroup of G G GGG through the natural diagonal embedding. Then, following the approach that I have developed in [12], (1.7) can be deduced from a certain simple trace formula for the "space" X = ( H , ξ ) G X = ( H , ξ ) ∖ G X=(H,xi)\\GX=(H, \xi) \backslash GX=(H,ξ)∖G. More precisely, this trace formula is roughly seeking to compute the trace of the convolution operators
ϕ L 2 ( X , ξ ) ( R ( f ) ϕ ) ( x ) = G f ( g ) ϕ ( x g ) d g , for f C c ( G ) Ï• ∈ L 2 ( X , ξ ) ↦ ( R ( f ) Ï• ) ( x ) = ∫ G   f ( g ) Ï• ( x g ) d g ,  for  f ∈ C c ∞ ( G ) phi inL^(2)(X,xi)|->(R(f)phi)(x)=int_(G)f(g)phi(xg)dg,quad" for "f inC_(c)^(oo)(G)\phi \in L^{2}(X, \xi) \mapsto(R(f) \phi)(x)=\int_{G} f(g) \phi(x g) d g, \quad \text { for } f \in C_{c}^{\infty}(G)ϕ∈L2(X,ξ)↦(R(f)Ï•)(x)=∫Gf(g)Ï•(xg)dg, for f∈Cc∞(G)
where L 2 ( X , ξ ) L 2 ( X , ξ ) L^(2)(X,xi)L^{2}(X, \xi)L2(X,ξ) denotes the Hilbert space of measurable functions ϕ Ï• phi\phiÏ• on G G GGG satisfying ϕ ( h g ) = ξ ( h ) ϕ ( g ) Ï• ( h g ) = ξ ( h ) Ï• ( g ) phi(hg)=xi(h)phi(g)\phi(h g)=\xi(h) \phi(g)Ï•(hg)=ξ(h)Ï•(g) for ( h , g ) H × G ( h , g ) ∈ H × G (h,g)in H xx G(h, g) \in H \times G(h,g)∈H×G and H G | ϕ ( x ) | 2 d x < ∫ H ∖ G   | Ï• ( x ) | 2 d x < ∞ int_(H\\G)|phi(x)|^(2)dx < oo\int_{H \backslash G}|\phi(x)|^{2} d x<\infty∫H∖G|Ï•(x)|2dx<∞. It is classical, and easy to see, that these operators are given by kernels,
( R ( f ) ϕ ) ( x ) = X K f ( x , y ) ϕ ( y ) d y , for ( f , ϕ ) C c ( G ) × L 2 ( X , ξ ) ( R ( f ) Ï• ) ( x ) = ∫ X   K f ( x , y ) Ï• ( y ) d y ,  for  ( f , Ï• ) ∈ C c ∞ ( G ) × L 2 ( X , ξ ) (R(f)phi)(x)=int_(X)K_(f)(x,y)phi(y)dy,quad" for "(f,phi)inC_(c)^(oo)(G)xxL^(2)(X,xi)(R(f) \phi)(x)=\int_{X} K_{f}(x, y) \phi(y) d y, \quad \text { for }(f, \phi) \in C_{c}^{\infty}(G) \times L^{2}(X, \xi)(R(f)Ï•)(x)=∫XKf(x,y)Ï•(y)dy, for (f,Ï•)∈Cc∞(G)×L2(X,ξ)
where K f ( x , y ) = H f ( x 1 h y ) ξ ( h ) d h K f ( x , y ) = ∫ H   f x − 1 h y ξ ( h ) d h K_(f)(x,y)=int_(H)f(x^(-1)hy)xi(h)dhK_{f}(x, y)=\int_{H} f\left(x^{-1} h y\right) \xi(h) d hKf(x,y)=∫Hf(x−1hy)ξ(h)dh. Thus, at a formal level (hence the quotation marks) we have
“Trace R ( f ) = X K f ( x , x ) d x "  “Trace  R ( f ) = ∫ X   K f ( x , x ) d x " " “Trace "R(f)=int_(X)K_(f)(x,x)dx"\text { “Trace } R(f)=\int_{X} K_{f}(x, x) d x " “Trace R(f)=∫XKf(x,x)dx"
However, neither of the two sides above make sense in general: the convolution operator is not of trace-class and the kernel not integrable over the diagonal. The basic idea is then to restrict oneself to a subspace of test functions for which at least one of the two expressions is meaningful. A convenient such subspace is that of strongly cuspidal functions introduced by Waldspurger in [47]: a function f C c ( G ) f ∈ C c ∞ ( G ) f inC_(c)^(oo)(G)f \in C_{c}^{\infty}(G)f∈Cc∞(G) is strongly cuspidal if for every proper parabolic subgroup P = M U G P = M U ⊊ G P=MU⊊GP=M U \subsetneq GP=MU⊊G, we have
U f ( m u ) d u = 0 , m M ∫ U   f ( m u ) d u = 0 , ∀ m ∈ M int_(U)f(mu)du=0,quad AA m in M\int_{U} f(m u) d u=0, \quad \forall m \in M∫Uf(mu)du=0,∀m∈M
Moreover, as is shown in [12], for f C c ( G ) f ∈ C c ∞ ( G ) f inC_(c)^(oo)(G)f \in C_{c}^{\infty}(G)f∈Cc∞(G) strongly cuspidal, the integral
J ( f ) = X K f ( x , x ) d x J ( f ) = ∫ X   K f ( x , x ) d x J(f)=int_(X)K_(f)(x,x)dxJ(f)=\int_{X} K_{f}(x, x) d xJ(f)=∫XKf(x,x)dx
is absolutely convergent (the argument of [12] is given in the context of Gan-Gross-Prasad for unitary groups but it can be adapted to a much more general context). Then, the aforementioned simple local trace formula expands the distribution f J ( f ) f → J ( f ) f rarr J(f)f \rightarrow J(f)f→J(f) in two different ways:
Theorem 1.2. For every strongly cuspidal f C c ( G ) f ∈ C c ∞ ( G ) f inC_(c)^(oo)(G)f \in C_{c}^{\infty}(G)f∈Cc∞(G), we have the identities
(1.12) Γ ( V , W ) r e g c f ( x ) d x = J ( f ) = X ( G ) m ( Π ) θ f ^ ( Π ) d Π (1.12) ∫ Γ ( V , W ) r e g   c f ( x ) d x = J ( f ) = ∫ X ( G )   m ( Π ) θ f ^ ( Π ) d Π {:(1.12)int_(Gamma(V,W))^(reg)c_(f)(x)dx=J(f)=int_(X(G))m(Pi) widehat(theta_(f))(Pi)d Pi:}\begin{equation*} \int_{\Gamma(V, W)}^{\mathrm{reg}} c_{f}(x) d x=J(f)=\int_{X(G)} m(\Pi) \widehat{\theta_{f}}(\Pi) d \Pi \tag{1.12} \end{equation*}(1.12)∫Γ(V,W)regcf(x)dx=J(f)=∫X(G)m(Π)θf^(Π)dΠ
where
  • c f ( x ) c f ( x ) c_(f)(x)c_{f}(x)cf(x) is the renormalized value of a function x θ f ( x ) x ↦ θ f ( x ) x|->theta_(f)(x)x \mapsto \theta_{f}(x)x↦θf(x) constructed from weighted orbital integrals of f f fff in the sense of Arthur [3] and whose local behavior is similar to that of Harish-Chandra characters on the group G;
  • X ( G ) ( G ) (G)(G)(G) is a certain space of virtual representations of G G GGG obtained by parabolic induction from the so-called elliptic representations (as defined in [6]) of Levi subgroups and f θ f ^ ( П ) f ↦ θ f ^ ( П ) f|-> widehat(theta_(f))(П)f \mapsto \widehat{\theta_{f}}(П)f↦θf^(П) is a weighted character in the sense of Arthur [4];
  • Finally, for an irreducible representation Π = π σ Π = Ï€ ⊗ σ Pi=pi ox sigma\Pi=\pi \otimes \sigmaΠ=π⊗σ of G , m ( Π ) G , m ( Π ) G,m(Pi)G, m(\Pi)G,m(Π) is defined as the multiplicity m ( π , σ ) m Ï€ , σ ∨ m(pi,sigma^(vv))m\left(\pi, \sigma^{\vee}\right)m(Ï€,σ∨) with σ σ ∨ sigma^(vv)\sigma^{\vee}σ∨ the smooth contragredient of σ σ sigma\sigmaσ.
We refer the reader to [12] for precise definitions of all the terms and a proof in the case of unitary groups. This was adapted in [36] to special orthogonal groups. The deduction of the integral formula (1.7) roughly goes as follows: we first show the multiplicity formula for representations that are properly parabolically induced by expressing both sides in terms of the inducing data and applying an induction hypothesis whereas for the remaining representations, the so-called elliptic representations, the formula can be obtained by applying the trace formula (1.12) to some sort of pseudocoefficient.
Finally, let us say a word on how the more refined part (2) of Conjecture 1.1 can be proven using this approach (so far it has only been done for p p ppp-adic groups in [49] and [9], following the previous slightly different method of Waldspurger, but there is little doubt that the techniques developed in [12] should allow to treat the case of real groups in a similar way). For Langlands parameters ϕ , ϕ Ï• , Ï• ′ phi,phi^(')\phi, \phi^{\prime}Ï•,ϕ′ as in Conjecture 1.1, as well as characters χ S ϕ ^ χ ∈ S Ï• ^ chi in widehat(S_(phi))\chi \in \widehat{S_{\phi}}χ∈SÏ•^, χ S ϕ ^ χ ′ ∈ S Ï• ′ ^ chi^(')in widehat(S_(phi^(')))\chi^{\prime} \in \widehat{S_{\phi^{\prime}}}χ′∈Sϕ′^, combining the multiplicity formula (1.7) with the general endoscopic character relations that characterize the Langlands correspondences for U ( V ) U ( V ) U(V)U(V)U(V) and U ( W ) U ( W ) U(W)U(W)U(W), we can express m ( π ( ϕ , χ ) , σ ( ϕ , χ ) ) m Ï€ ( Ï• , χ ) , σ Ï• ′ , χ ′ m(pi(phi,chi),sigma(phi^('),chi^(')))m\left(\pi(\phi, \chi), \sigma\left(\phi^{\prime}, \chi^{\prime}\right)\right)m(Ï€(Ï•,χ),σ(ϕ′,χ′)) as a sum of integrals of (renormalized) twisted characters on some products G L n ( E ) × G L m ( E ) G L n ( E ) × G L m ( E ) GL_(n)(E)xxGL_(m)(E)\mathrm{GL}_{n}(E) \times \mathrm{GL}_{m}(E)GLn(E)×GLm(E). The remaining ingredient is to relate these integrals of twisted characters to the epsilon factors of pairs defined by Jacquet-Piatetski-Shapiro-Shalika in [31]. More precisely, these expressions involve the twisted characters of tempered irreducible representations π G L , σ G L Ï€ G L , σ G L pi^(GL),sigma^(GL)\pi^{\mathrm{GL}}, \sigma^{\mathrm{GL}}Ï€GL,σGL of general linear groups G L n ( E ) , G L m ( E ) G L n ( E ) , G L m ( E ) GL_(n)(E),GL_(m)(E)\mathrm{GL}_{n}(E), \mathrm{GL}_{m}(E)GLn(E),GLm(E), with n m n ⩾ m n >= mn \geqslant mn⩾m of distinct parities, which are self-dual (in the orthogonal case) or conjugate-self-dual (in the unitary case). These properties of (conjugate-)self-duality imply that π G L Ï€ G L pi^(GL)\pi^{\mathrm{GL}}Ï€GL and σ G L σ G L sigma^(GL)\sigma^{\mathrm{GL}}σGL extend to representations π G L , σ G L Ï€ G L , σ G L pi^(GL),sigma^(GL)\pi^{\mathrm{GL}}, \sigma^{\mathrm{GL}}Ï€GL,σGL of the nonconnected groups G L n ( E ) θ n G L n ( E ) ⋊ θ n GL_(n)(E)><|(:theta_(n):)\mathrm{GL}_{n}(E) \rtimes\left\langle\theta_{n}\right\rangleGLn(E)⋊⟨θn⟩ and G L m ( E ) θ m G L m ( E ) ⋊ θ m GL_(m)(E)><|(:theta_(m):)\mathrm{GL}_{m}(E) \rtimes\left\langle\theta_{m}\right\rangleGLm(E)⋊⟨θm⟩, respectively, where θ k ( k = n , m ) θ k ( k = n , m ) theta_(k)(k=n,m)\theta_{k}(k=n, m)θk(k=n,m) denotes the automorphism g t ( g c ) 1 g ↦ t g c − 1 g|->^(t)(g^(c))^(-1)g \mapsto^{t}\left(g^{c}\right)^{-1}g↦t(gc)−1. The twisted characters in question are then the restrictions Θ π G L Θ Ï€ G L Theta_(pi^(GL))\Theta_{\pi^{\mathrm{GL}}}ΘπGL and Θ σ G L Θ σ G L Theta_(sigma^(GL))\Theta_{\sigma^{\mathrm{GL}}}ΘσGL of the Harish-Chandra characters of π G L Ï€ G L pi^(GL)\pi^{\mathrm{GL}}Ï€GL and σ G L σ G L sigma^(GL)\sigma^{\mathrm{GL}}σGL to the nonneutral components G L n ~ ( E ) = G L n ( E ) θ n G L n ~ ( E ) = G L n ( E ) θ n widetilde(GL_(n))(E)=GL_(n)(E)theta_(n)\widetilde{\mathrm{GL}_{n}}(E)=\mathrm{GL}_{n}(E) \theta_{n}GLn~(E)=GLn(E)θn and G L m ~ ( E ) = G L m ~ ( E ) = widetilde(GL_(m))(E)=\widetilde{\mathrm{GL}_{m}}(E)=GLm~(E)= G L m ( E ) θ m G L m ( E ) θ m GL_(m)(E)theta_(m)\mathrm{GL}_{m}(E) \theta_{m}GLm(E)θm, respectively. Replacing the functions c π , c σ c Ï€ , c σ c_(pi),c_(sigma)c_{\pi}, c_{\sigma}cÏ€,cσ by similar suitable renormalizations of these twisted characters at singular semisimple conjugacy classes, there is a formula very analogous to (1.7) for the ε ε epsi\varepsilonε-factor of pair ε ( π G L × σ G L , ψ ) ε Ï€ G L × σ G L , ψ epsi(pi^(GL)xxsigma^(GL),psi)\varepsilon\left(\pi^{\mathrm{GL}} \times \sigma^{\mathrm{GL}}, \psi\right)ε(Ï€GL×σGL,ψ).
For p p ppp-adic fields, this formula was established in [48] in the self-dual case and in [8] in the conjugate-self-dual case. The proof follows closely that of (1.7) and is based on a local trace formula very similar to that of Theorem 1.2 for the natural action of G := G L n ~ ( E ) × G ′ := G L n ~ ( E ) × G^('):= widetilde(GL_(n))(E)xxG^{\prime}:=\widetilde{\mathrm{GL}_{n}}(E) \timesG′:=GLn~(E)× G L m ~ ( E ) G L m ~ ( E ) widetilde(GL_(m))(E)\widetilde{\mathrm{GL}_{m}}(E)GLm~(E) on the homogeneous space X = H G X ′ = H ′ ∖ G ′ X^(')=H^(')\\G^(')X^{\prime}=H^{\prime} \backslash G^{\prime}X′=H′∖G′ where G = GL n ( E ) × G L m ( E ) G ′ = GL n ⁡ ( E ) × G L m ( E ) G^(')=GL_(n)(E)xxGL_(m)(E)G^{\prime}=\operatorname{GL}_{n}(E) \times \mathrm{GL}_{m}(E)G′=GLn⁡(E)×GLm(E) and H = GL m ( E ) N H ′ = GL m ⁡ ( E ) ⋉ N ′ H^(')=GL_(m)(E)|><N^(')H^{\prime}=\operatorname{GL}_{m}(E) \ltimes N^{\prime}H′=GLm⁡(E)⋉N′ is the semidirect product with a unipotent subgroup N N ′ N^(')N^{\prime}N′ whose definition is analogous to that of N N NNN. More precisely, there is also a similar unitary character ξ ξ ′ xi^(')\xi^{\prime}ξ′ of N N ′ N^(')N^{\prime}N′ that is G L m ( E ) G L m ( E ) GL_(m)(E)\mathrm{GL}_{m}(E)GLm(E)-invariant and the twisted trace formula we are mentioning is roughly trying to compute the trace of convolution operators R ( f ) R ( f ) R(f)R(f)R(f) of functions f C c ( G ) f ∈ C c ∞ G ′ f inC_(c)^(oo)(G^('))f \in C_{c}^{\infty}\left(G^{\prime}\right)f∈Cc∞(G′) on
L 2 ( X , ξ ) L 2 X ′ , ξ ′ L^(2)(X^('),xi^('))L^{2}\left(X^{\prime}, \xi^{\prime}\right)L2(X′,ξ′). Rather than describing it in details, let us just explain how the ε ε epsi\varepsilonε-factors show up in the analysis. As in Theorem 1.2, one of the main ingredient on the spectral side of this trace formula is a twisted multiplicity m ( π G L σ G L ) m Ï€ G L ⊗ σ G L m(pi^(GL)oxsigma^(GL))m\left(\pi^{\mathrm{GL}} \otimes \sigma^{\mathrm{GL}}\right)m(Ï€GL⊗σGL) which computes the trace of a natural operator on the space of intertwiners
(1.13) Hom H ( π G L σ G L , ξ ) (1.13) Hom H ⁡ Ï€ G L ⊗ σ G L , ξ {:(1.13)Hom_(H)(pi^(GL)oxsigma^(GL),xi):}\begin{equation*} \operatorname{Hom}_{H}\left(\pi^{\mathrm{GL}} \otimes \sigma^{\mathrm{GL}}, \xi\right) \tag{1.13} \end{equation*}(1.13)HomH⁡(Ï€GL⊗σGL,ξ)
The operator in question is given by ( π G L σ G L ) ( θ ) â„“ ↦ â„“ ∘ Ï€ G L ⊗ σ G L ( θ ) â„“|->â„“@(pi^(GL)oxsigma^(GL))(theta)\ell \mapsto \ell \circ\left(\pi^{\mathrm{GL}} \otimes \sigma^{\mathrm{GL}}\right)(\theta)ℓ↦ℓ∘(Ï€GL⊗σGL)(θ) where θ G L ~ n ( E ) × G L ~ m ( E ) θ ∈ G L ~ n ( E ) × G L ~ m ( E ) theta in widetilde(GL)_(n)(E)xx widetilde(GL)_(m)(E)\theta \in \widetilde{\mathrm{GL}}_{n}(E) \times \widetilde{\mathrm{GL}}_{m}(E)θ∈GL~n(E)×GL~m(E) is a certain element stabilizing the pair ( H , ξ ) ( H , ξ ) (H,xi)(H, \xi)(H,ξ) (which is anyway needed to extend the right action of G G ′ G^(')G^{\prime}G′ on L 2 ( X , ξ ) L 2 X ′ , ξ ′ L^(2)(X^('),xi^('))L^{2}\left(X^{\prime}, \xi^{\prime}\right)L2(X′,ξ′) to an action of G ) G ′ {:G^('))\left.G^{\prime}\right)G′). Actually, it turns out that the space (1.13) is always one-dimensional and a reformulation of the so-called local functional equation from [31] shows that this operator is essentially acting (for suitable normalizations of π G L , σ G L Ï€ G L , σ G L pi^(GL),sigma^(GL)\pi^{\mathrm{GL}}, \sigma^{\mathrm{GL}}Ï€GL,σGL and up to more elementary factors) as multiplication by the ε ε epsi\varepsilonε-factor ε ( π × σ , ψ ) ε ( Ï€ × σ , ψ ) epsi(pi xx sigma,psi)\varepsilon(\pi \times \sigma, \psi)ε(π×σ,ψ).

2. THE GLOBAL GAN-GROSS-PRASAD CONJECTURES AND ICHINO-IKEDA REFINEMENTS

2.1. Statements and results

We now move to a global setting. Let E / F E / F E//FE / FE/F be a quadratic extension of number fields and W V W ⊂ V W sub VW \subset VW⊂V be Hermitian spaces over E E EEE satisfying condition (1.1) (there are similar, and actually prior, conjectures for orthogonal groups, but here we will concentrate on unitary groups for which much more is known). By a construction similar to that from the previous section, we may obtain from these data a triple ( G , H , ξ ) ( G , H , ξ ) (G,H,xi)(G, H, \xi)(G,H,ξ) where G = U ( V ) × U ( W ) G = U ( V ) × U ( W ) G=U(V)xx U(W)G=U(V) \times U(W)G=U(V)×U(W), H = U ( W ) N H = U ( W ) ⋉ N H=U(W)|><NH=U(W) \ltimes NH=U(W)⋉N is a subgroup of G G GGG (which we will this time consider as algebraic groups over F F FFF ) and ξ : N ( A F ) C × Î¾ : N A F → C × xi:N(A_(F))rarrC^(xx)\xi: N\left(\mathbb{A}_{F}\right) \rightarrow \mathbb{C}^{\times}ξ:N(AF)→C×is a character on the adelic points of N N NNN trivial on the subgroup N ( F ) N ( F ) N(F)N(F)N(F) and that extends to a character of H ( A F ) H A F H(A_(F))H\left(\mathbb{A}_{F}\right)H(AF) trivial on U ( W ) ( A F ) U ( W ) A F U(W)(A_(F))U(W)\left(\mathbb{A}_{F}\right)U(W)(AF).
The global analog of the previous branching problem is that of characterizing the nonvanishing of the automorphic period associated to the pair ( H , ξ ) ( H , ξ ) (H,xi)(H, \xi)(H,ξ). More precisely, if π = π V π W A cusp ( G ( F ) G ( A F ) ) Ï€ = Ï€ V ⊗ Ï€ W ↪ A cusp  G ( F ) ∖ G A F pi=pi_(V)oxpi_(W)↪A_("cusp ")(G(F)\\G(A_(F)))\pi=\pi_{V} \otimes \pi_{W} \hookrightarrow \mathcal{A}_{\text {cusp }}\left(G(F) \backslash G\left(\mathbb{A}_{F}\right)\right)Ï€=Ï€V⊗πW↪Acusp (G(F)∖G(AF)) is a cuspidal automorphic representation of G ( A F ) G A F G(A_(F))G\left(\mathbb{A}_{F}\right)G(AF), we consider the automorphic period
P H , ξ : π C (2.1) P H , ξ ( φ ) = [ H ] φ ( h ) ξ ( h ) d h P H , ξ : Ï€ → C (2.1) P H , ξ ( φ ) = ∫ [ H ]   φ ( h ) ξ ( h ) d h {:[P_(H,xi):pi rarrC],[(2.1)P_(H,xi)(varphi)=int_([H])varphi(h)xi(h)dh]:}\begin{align*} \mathcal{P}_{H, \xi} & : \pi \rightarrow \mathbb{C} \\ \mathcal{P}_{H, \xi}(\varphi) & =\int_{[H]} \varphi(h) \xi(h) d h \tag{2.1} \end{align*}PH,ξ:π→C(2.1)PH,ξ(φ)=∫[H]φ(h)ξ(h)dh
where here and throughout the rest of the paper, for a linear algebraic group R R RRR over F F FFF, we denote by [ R ] = R ( F ) R ( A F ) [ R ] = R ( F ) ∖ R A F [R]=R(F)\\R(A_(F))[R]=R(F) \backslash R\left(\mathbb{A}_{F}\right)[R]=R(F)∖R(AF) the corresponding automorphic quotient. On the other hand, let π E = π V , E π W , E Ï€ E = Ï€ V , E ⊗ Ï€ W , E pi_(E)=pi_(V,E)oxpi_(W,E)\pi_{E}=\pi_{V, E} \otimes \pi_{W, E}Ï€E=Ï€V,E⊗πW,E be the (weak) base-change of π Ï€ pi\piÏ€ to G L n ( A E ) × G L m ( A E ) G L n A E × G L m A E GL_(n)(A_(E))xxGL_(m)(A_(E))\mathrm{GL}_{n}\left(\mathbb{A}_{E}\right) \times \mathrm{GL}_{m}\left(\mathbb{A}_{E}\right)GLn(AE)×GLm(AE) where ( n , m ) = ( dim ( V ) , dim ( W ) ) ( n , m ) = ( dim ⁡ ( V ) , dim ⁡ ( W ) ) (n,m)=(dim(V),dim(W))(n, m)=(\operatorname{dim}(V), \operatorname{dim}(W))(n,m)=(dim⁡(V),dim⁡(W)). Here, π V , E , π W , E Ï€ V , E , Ï€ W , E pi_(V,E),pi_(W,E)\pi_{V, E}, \pi_{W, E}Ï€V,E,Ï€W,E are automorphic representations whose Satake parameters at almost every unramified places are the image by the base-change homomorphisms L U ( V ) L Res E / F G L n , E , L U ( W ) L Res E / F G L m , E ( L U ( V ) → L Res E / F ⁡ G L n , E , L U ( W ) → L Res E / F ⁡ G L m , E ^(L)U(V)rarr^(L)Res_(E//F)GL_(n,E),^(L)U(W)rarr^(L)Res_(E//F)GL_(m,E)(:}{ }^{L} U(V) \rightarrow{ }^{L} \operatorname{Res}_{E / F} \mathrm{GL}_{n, E},{ }^{L} U(W) \rightarrow{ }^{L} \operatorname{Res}_{E / F} \mathrm{GL}_{m, E}\left(\right.LU(V)→LResE/F⁡GLn,E,LU(W)→LResE/F⁡GLm,E( where Res E / F Res E / F Res_(E//F)\operatorname{Res}_{E / F}ResE/F denotes Weil's restriction of scalars) of the local Satake parameters of π V , π W Ï€ V , Ï€ W pi_(V),pi_(W)\pi_{V}, \pi_{W}Ï€V,Ï€W, respectively. The existence of these weak base-changes in general is one of the main results of [34,39]. Also, although π V , E , π W , E Ï€ V , E , Ï€ W , E pi_(V,E),pi_(W,E)\pi_{V, E}, \pi_{W, E}Ï€V,E,Ï€W,E are not always cuspidal, they are isobaric sums of cuspidal representations which implies, by a result of Jacquet and Shalika, that they are uniquely
determined by their Satake parameters at almost all places hence that the weak base-change π E Ï€ E pi_(E)\pi_{E}Ï€E is unique. We denote by
L ( s , π E ) = L ( s , π V , E × π W , E ) L s , Ï€ E = L s , Ï€ V , E × Ï€ W , E L(s,pi_(E))=L(s,pi_(V,E)xxpi_(W,E))L\left(s, \pi_{E}\right)=L\left(s, \pi_{V, E} \times \pi_{W, E}\right)L(s,Ï€E)=L(s,Ï€V,E×πW,E)
the corresponding completed Rankin-Selberg L L LLL-function associated to π V , E Ï€ V , E pi_(V,E)\pi_{V, E}Ï€V,E and π W , E Ï€ W , E pi_(W,E)\pi_{W, E}Ï€W,E.
Define the automorphic L L LLL-packet of π Ï€ pi\piÏ€ as the set of cuspidal automorphic representations π Ï€ ′ pi^(')\pi^{\prime}π′ of the various pure inner form G = U ( V ) × U ( W ) G ′ = U V ′ × U W ′ G^(')=U(V^('))xx U(W^('))G^{\prime}=U\left(V^{\prime}\right) \times U\left(W^{\prime}\right)G′=U(V′)×U(W′) of G G GGG with the same base-change π E = π E Ï€ E ′ = Ï€ E pi_(E)^(')=pi_(E)\pi_{E}^{\prime}=\pi_{E}Ï€E′=Ï€E as π Ï€ pi\piÏ€. By the Jacquet-Shalika theorem again and injectivity of basechange homomorphisms at the level of conjugacy classes, it is equivalent to asking that π Ï€ pi\piÏ€ and π Ï€ ′ pi^(')\pi^{\prime}π′ are nearly equivalent, that is, π v π v Ï€ v ≃ Ï€ v ′ pi_(v)≃pi_(v)^(')\pi_{v} \simeq \pi_{v}^{\prime}Ï€v≃πv′ for almost all places v v vvv (this makes sense since G v G v G v ≃ G v ′ G_(v)≃G_(v)^(')G_{v} \simeq G_{v}^{\prime}Gv≃Gv′ for almost all v v vvv ). Moreover, for a relevant pure inner form G G ′ G^(')G^{\prime}G′ of G G GGG, we can define a pair ( H , ξ ) H ′ , ξ ′ (H^('),xi^('))\left(H^{\prime}, \xi^{\prime}\right)(H′,ξ′) in exactly the same way as ( H , ξ ) ( H , ξ ) (H,xi)(H, \xi)(H,ξ). The global version of the Gan-Gross-Prasad conjecture can now be stated as follows:
Conjecture 2.1 (Gan-Gross-Prasad [23]). Assume that π E Ï€ E pi_(E)\pi_{E}Ï€E is generic. Then, the following assertions are equivalent:
(1) L ( 1 2 , π E ) 0 L 1 2 , Ï€ E ≠ 0 L((1)/(2),pi_(E))!=0L\left(\frac{1}{2}, \pi_{E}\right) \neq 0L(12,Ï€E)≠0;
(2) There exists a relevant pure inner form G = U ( W ) × U ( V ) G ′ = U W ′ × U V ′ G^(')=U(W^('))xx U(V^('))G^{\prime}=U\left(W^{\prime}\right) \times U\left(V^{\prime}\right)G′=U(W′)×U(V′) of G G GGG (see Section 1.1 for the definition of a relevant pure inner form), a cuspidal automorphic representation π Ï€ ′ pi^(')\pi^{\prime}π′ of G ( A F ) G ′ A F G^(')(A_(F))G^{\prime}\left(\mathbb{A}_{F}\right)G′(AF) in the same automorphic L L LLL-packet as π Ï€ pi\piÏ€ and a form φ π φ ′ ∈ Ï€ ′ varphi^(')inpi^(')\varphi^{\prime} \in \pi^{\prime}φ′∈π′ such that
P H , ξ ( φ ) 0 P H ′ , ξ ′ φ ′ ≠ 0 P_(H^('),xi^('))(varphi^('))!=0\mathscr{P}_{H^{\prime}, \xi^{\prime}}\left(\varphi^{\prime}\right) \neq 0PH′,ξ′(φ′)≠0
When ( dim ( V ) , dim ( W ) ) = ( 2 , 1 ) ( dim ⁡ ( V ) , dim ⁡ ( W ) ) = ( 2 , 1 ) (dim(V),dim(W))=(2,1)(\operatorname{dim}(V), \operatorname{dim}(W))=(2,1)(dim⁡(V),dim⁡(W))=(2,1), the conjecture essentially reduces to the celebrated theorem of Waldspurger [46] on toric periods for G L 2 G L 2 GL_(2)\mathrm{GL}_{2}GL2. Actually, as explained in the introduction, Waldspurger's result is more precise and gives an explicit identity relating (the square of) P H ( φ ) P H ( φ ) P_(H)(varphi)\mathcal{P}_{H}(\varphi)PH(φ) to the central value L ( 1 2 , π E ) L 1 2 , Ï€ E L((1)/(2),pi_(E))L\left(\frac{1}{2}, \pi_{E}\right)L(12,Ï€E).
There is also a similar conjecture for special orthogonal groups which actually predates the one for unitary groups [26] (as well as other conjectures for the so-called FourierJacobi periods on unitary and symplectic/metaplectic groups stated in [23]). In [30], Ichino and Ikeda have proposed a refinement of this conjecture for S O ( n ) × SO ( n 1 ) S O ( n ) × SO ⁡ ( n − 1 ) SO(n)xx SO(n-1)\mathrm{SO}(n) \times \operatorname{SO}(n-1)SO(n)×SO⁡(n−1) in the form of a precise identity generalizing Waldspurger's formula. Subsequently, similar refinements have been proposed by R. N. Harris [27], for U ( n ) × U ( n 1 ) U ( n ) × U ( n − 1 ) U(n)xx U(n-1)U(n) \times U(n-1)U(n)×U(n−1), and then by Y. Liu [35] for general Bessel periods on orthogonal or unitary groups.
In order to state this refinement, we need to introduce two extra ingredients, namely local periods and a certain finite group S π S Ï€ S_(pi)S_{\pi}SÏ€ of endoscopic nature.
We start with the local periods. We endow H ( A F ) H A F H(A_(F))H\left(\mathbb{A}_{F}\right)H(AF) with its global Tamagawa measure d h d h dhd hdh (this is the measure with which we will normalize the period integral (2.1)) and we fix a factorization d h = v d h v d h = ∏ v   d h v dh=prod_(v)dh_(v)d h=\prod_{v} d h_{v}dh=∏vdhv into a product of local Haar measures. We also fix a decomposition π = v π v Ï€ = ⨂ v ′   Ï€ v pi=⨂_(v)^(')pi_(v)\pi=\bigotimes_{v}^{\prime} \pi_{v}Ï€=⨂v′πv of π Ï€ pi\piÏ€ into smooth irreducible representations of the localizations
G v = G ( F v ) G v = G F v G_(v)=G(F_(v))G_{v}=G\left(F_{v}\right)Gv=G(Fv) as well as a factorization , P e t = v , v ⟨ ⋅ , ⋅ ⟩ P e t = ∏ v   ⟨ ⋅ , ⋅ ⟩ v (:*,*:)_(Pet)=prod_(v)(:*,*:)_(v)\langle\cdot, \cdot\rangle_{\mathrm{Pet}}=\prod_{v}\langle\cdot, \cdot\rangle_{v}⟨⋅,⋅⟩Pet=∏v⟨⋅,⋅⟩v of the Petersson inner product
φ , φ P e t = G ( F ) G ( A F ) | φ ( g ) | 2 d g ⟨ φ , φ ⟩ P e t = ∫ G ( F ) ∖ G A F   | φ ( g ) | 2 d g (:varphi,varphi:)_(Pet)=int_(G(F)\\G(A_(F)))|varphi(g)|^(2)dg\langle\varphi, \varphi\rangle_{\mathrm{Pet}}=\int_{G(F) \backslash G\left(\mathbb{A}_{F}\right)}|\varphi(g)|^{2} d g⟨φ,φ⟩Pet=∫G(F)∖G(AF)|φ(g)|2dg
(which we also normalize using the Tamagawa measure on G ( A F ) G A F G(A_(F))G\left(\mathbb{A}_{F}\right)G(AF) ) into local invariant inner products. The local periods are now given by the sesquilinear forms
(2.2) P H , ξ , v : φ v φ v π v π v H v r e g π v ( h v ) φ v , φ v v ξ v ( h v ) d h v (2.2) P H , ξ , v : φ v ⊗ φ v ′ ∈ Ï€ v ⊗ Ï€ v ↦ ∫ H v r e g   Ï€ v h v φ v , φ v ′ v ξ v h v d h v {:(2.2)P_(H,xi,v):varphi_(v)oxvarphi_(v)^(')inpi_(v)oxpi_(v)|->int_(H_(v))^(reg)(:pi_(v)(h_(v))varphi_(v),varphi_(v)^('):)_(v)xi_(v)(h_(v))dh_(v):}\begin{equation*} \mathcal{P}_{H, \xi, v}: \varphi_{v} \otimes \varphi_{v}^{\prime} \in \pi_{v} \otimes \pi_{v} \mapsto \int_{H_{v}}^{\mathrm{reg}}\left\langle\pi_{v}\left(h_{v}\right) \varphi_{v}, \varphi_{v}^{\prime}\right\rangle_{v} \xi_{v}\left(h_{v}\right) d h_{v} \tag{2.2} \end{equation*}(2.2)PH,ξ,v:φv⊗φv′∈πv⊗πv↦∫Hvreg⟨πv(hv)φv,φv′⟩vξv(hv)dhv
The above integral of matrix coefficient is actually not convergent in general and has to be regularized (hence the "reg" sign above the integral). This regularization is, moreover, only possible under the extra assumption that the local component π v Ï€ v pi_(v)\pi_{v}Ï€v is tempered. It is expected (under the Generalized Ramanujan Conjecture) that the hypothesis of the base-change π E Ï€ E pi_(E)\pi_{E}Ï€E being generic implies that each of the local component π v Ï€ v pi_(v)\pi_{v}Ï€v is tempered, but this is far from being known in general. Assuming now that π v Ï€ v pi_(v)\pi_{v}Ï€v is tempered at every place v v vvv, an unramified computation shows that for almost all places v v vvv, if φ v π v G ( O v ) φ v ∈ Ï€ v G O v varphi_(v)inpi_(v)^(G(O_(v)))\varphi_{v} \in \pi_{v}^{G\left(\mathcal{O}_{v}\right)}φv∈πvG(Ov) is a spherical vector such that φ v , φ v v = 1 φ v , φ v v = 1 (:varphi_(v),varphi_(v):)_(v)=1\left\langle\varphi_{v}, \varphi_{v}\right\rangle_{v}=1⟨φv,φv⟩v=1, we have
P H , ξ , v ( φ v , φ v ) = Δ v L ( 1 2 , π E , v ) L ( 1 , π v , A d ) P H , ξ , v φ v , φ v = Δ v L 1 2 , Ï€ E , v L 1 , Ï€ v , A d P_(H,xi,v)(varphi_(v),varphi_(v))=Delta_(v)(L((1)/(2),pi_(E,v)))/(L(1,pi_(v),Ad))\mathcal{P}_{H, \xi, v}\left(\varphi_{v}, \varphi_{v}\right)=\Delta_{v} \frac{L\left(\frac{1}{2}, \pi_{E, v}\right)}{L\left(1, \pi_{v}, A d\right)}PH,ξ,v(φv,φv)=ΔvL(12,Ï€E,v)L(1,Ï€v,Ad)
where L ( 1 2 , π E , v ) , L ( 1 , π v , A d ) L 1 2 , Ï€ E , v , L 1 , Ï€ v , A d L((1)/(2),pi_(E,v)),L(1,pi_(v),Ad)L\left(\frac{1}{2}, \pi_{E, v}\right), L\left(1, \pi_{v}, A d\right)L(12,Ï€E,v),L(1,Ï€v,Ad) denote the local Rankin-Selberg and adjoint L L LLL-factors of π E Ï€ E pi_(E)\pi_{E}Ï€E and π Ï€ pi\piÏ€, respectively, whereas Δ v Δ v Delta_(v)\Delta_{v}Δv stands for the product of local abelian L L LLL-factors
Δ v = i = 1 n L ( i , η E v / F v i ) Δ v = ∏ i = 1 n   L i , η E v / F v i Delta_(v)=prod_(i=1)^(n)L(i,eta_(E_(v)//F_(v))^(i))\Delta_{v}=\prod_{i=1}^{n} L\left(i, \eta_{E_{v} / F_{v}}^{i}\right)Δv=∏i=1nL(i,ηEv/Fvi)
with η E v / F v η E v / F v eta_(E_(v)//F_(v))\eta_{E_{v} / F_{v}}ηEv/Fv the quadratic character associated to the local extension E v / F v E v / F v E_(v)//F_(v)E_{v} / F_{v}Ev/Fv and n = dim ( V ) n = dim ⁡ ( V ) n=dim(V)n=\operatorname{dim}(V)n=dim⁡(V). The normalized local periods are then defined by
P H , ξ , v ( φ v , φ v ) = Δ v 1 L ( 1 , π v , A d ) L ( 1 2 , π E , v ) P H , ξ , v ( φ v , φ v ) P H , ξ , v â™® φ v , φ v = Δ v − 1 L 1 , Ï€ v , A d L 1 2 , Ï€ E , v P H , ξ , v φ v , φ v P_(H,xi,v)^(â™®)(varphi_(v),varphi_(v))=Delta_(v)^(-1)(L(1,pi_(v),Ad))/(L((1)/(2),pi_(E,v)))P_(H,xi,v)(varphi_(v),varphi_(v))\mathcal{P}_{H, \xi, v}^{\natural}\left(\varphi_{v}, \varphi_{v}\right)=\Delta_{v}^{-1} \frac{L\left(1, \pi_{v}, A d\right)}{L\left(\frac{1}{2}, \pi_{E, v}\right)} \mathcal{P}_{H, \xi, v}\left(\varphi_{v}, \varphi_{v}\right)PH,ξ,vâ™®(φv,φv)=Δv−1L(1,Ï€v,Ad)L(12,Ï€E,v)PH,ξ,v(φv,φv)
Finally, writing the base-change π V E Ï€ V E pi_(V_(E))\pi_{V_{E}}Ï€VE and π W , E Ï€ W , E pi_(W,E)\pi_{W, E}Ï€W,E as isobaric sums
π V , E = π V , 1 π V , k , π W , E = π W , 1 π W , l Ï€ V , E = Ï€ V , 1 ⊞ ⋯ ⊞ Ï€ V , k , Ï€ W , E = Ï€ W , 1 ⊞ ⋯ ⊞ Ï€ W , l pi_(V,E)=pi_(V,1)⊞cdots⊞pi_(V,k),quadpi_(W,E)=pi_(W,1)⊞cdots⊞pi_(W,l)\pi_{V, E}=\pi_{V, 1} \boxplus \cdots \boxplus \pi_{V, k}, \quad \pi_{W, E}=\pi_{W, 1} \boxplus \cdots \boxplus \pi_{W, l}Ï€V,E=Ï€V,1⊞⋯⊞πV,k,Ï€W,E=Ï€W,1⊞⋯⊞πW,l
of cuspidal automorphic representations of some general linear groups, we set S π = ( Z / 2 Z ) k + l S Ï€ = ( Z / 2 Z ) k + l S_(pi)=(Z//2Z)^(k+l)S_{\pi}=(\mathbb{Z} / 2 \mathbb{Z})^{k+l}SÏ€=(Z/2Z)k+l. It serves as a substitute for the centralizer of the, yet nonexistent in general, global Langlands parameter of π Ï€ pi\piÏ€.
Conjecture 2.2 (Ichino-Ikeda, N. Harris, Y. Liu). Assume that for every place v v vvv of F , π v F , Ï€ v F,pi_(v)F, \pi_{v}F,Ï€v is a tempered representation. Then, for every factorizable vector φ = v φ v π φ = ⊗ v ′ φ v ∈ Ï€ varphi=ox_(v)^(')varphi_(v)in pi\varphi=\otimes_{v}^{\prime} \varphi_{v} \in \piφ=⊗v′φv∈π, we have
(2.3) | P H , ξ ( φ ) | 2 = | S π | 1 Δ L ( 1 2 , π E ) L ( 1 , π , A d ) v P H , ξ , v ( φ v , φ v ) (2.3) P H , ξ ( φ ) 2 = S Ï€ − 1 Δ L 1 2 , Ï€ E L ( 1 , Ï€ , A d ) ∏ v   P H , ξ , v â™® φ v , φ v {:(2.3)|P_(H,xi)(varphi)|^(2)=|S_(pi)|^(-1)Delta(L((1)/(2),pi_(E)))/(L(1,pi,Ad))prod_(v)P_(H,xi,v)^(â™®)(varphi_(v),varphi_(v)):}\begin{equation*} \left|\mathcal{P}_{H, \xi}(\varphi)\right|^{2}=\left|S_{\pi}\right|^{-1} \Delta \frac{L\left(\frac{1}{2}, \pi_{E}\right)}{L(1, \pi, A d)} \prod_{v} \mathcal{P}_{H, \xi, v}^{\natural}\left(\varphi_{v}, \varphi_{v}\right) \tag{2.3} \end{equation*}